Calculus/Sequences and Series/Exercises

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The following exercises test your understanding of infinite sequences and series. You may want to review that material before trying these problems.

Each question is followed by a "Hint" (usually a quick indication of the most efficient way to work the problem), the "Answer only" (what it sounds like), and finally a "Full solution" (showing all the steps required to get to the right answer). These should show up as "collapsed" or "hidden" sections (click on the title to display the contents), but some older web browsers might not be able to display them correctly (i.e., showing the content when it should be hidden). If this is true for your browser (or if you're looking at a printed version), you should take care not to "see too much" before you start thinking of how to work each problem.

Sequences[edit]

Consider the infinite sequence

a_n=n/2^n,\ n=1,\ldots,\infty.
  • Is the sequence monotonically increasing or decreasing?
Hint

compare adjacent terms algebraically or take a derivative

Answer only

monotonically decreasing (strictly decreasing starting with the second term)

Full solution

One may either consider how a_{n+1} compares to a_n algebraically and try to show that one is greater than the other, or take the derivative of a_n with respect to n and check where it is positive or negative.

Algebraically, since

a_{n+1}-a_n=\frac{n+1}{2^{n+1}}-\frac{n}{2^n}=\frac{n+1}{2^{n+1}}-\frac{2n}{2^{n+1}}=\frac{1-n}{2^{n+1}},

we see that a_{n+1}<a_n for n>1. That is, starting from the second term, the sequence is strictly decreasing. It is easy to check how the first two terms compare by just plugging in n=1 and n=2:

a_1=1/2
a_2=2/4=1/2

The first two terms are equal and thereafter the terms are strictly decreasing. Therefore, the sequence is monotonically decreasing.

Using calculus,

\frac{d}{dn}\,\frac{n}{2^n}=\frac{2^n\cdot1-n\cdot2^n\ln2}{(2^n)^2}=\frac{2^n(1-n\ln2)}{2^{2n}}=\frac{1-n\ln2}{2^n},

which is negative for n>1/\ln2\approx1.44>1. The rest of the argument is the same as before.

  • Is the sequence bounded from below, from above, both, or neither?
Hint

consider what kind of values are taken on by the numerator and denominator, and use the previous answer

Answer only

bounded from below and from above (both)

Full solution

The sequence is bounded from below because the terms are clearly positive (greater than 0) for all values of n. Also, since the sequence is decreasing (see the previous problem), the maximum value of the sequence must be the value of the first term. So the sequence is bounded from above (by the value 1/2), as well.

  • Does the sequence converge or diverge?
Hint

use the previous two answers to make a conclusion, or take a limit

Answer only

converges

Full solution

By the previous two answers, the sequence is bounded from below and monotonically decreasing, thus by a theorem it must converge.

To show this directly, consider the limit

\lim_{n\to\infty} \frac{n}{2^n}=\lim_{n\to\infty} \frac{1}{2^n\ln2}=0.

The two limits are equal by L'Hôpital's Rule, since the numerator and denominator of the expression in the first limit both grow to infinity.

Since the limit exists, it is the number to which the sequence converges.

Partial sums[edit]

Assume that the nth partial sum of a series is given by s_n=2-\frac{1}{3^n}.

  • Does the series converge? If so, to what value?
Hint

take a limit

Answer only

converges to 2

Full solution

The series converges to 2 since

s=\lim_{n\to\infty} s_n=\lim_{n\to\infty} \left(2-\frac{1}{3^n}\right)=2.
  • What is the formula for the nth term of the series?
Show hint

s_n=a_1+\cdots+a_{n-1}+a_n=s_{n-1}+a_n

Show answer only

a_n=\frac{2}{3^n}

Show full solution

a_n=s_n-s_{n-1}=\left(2-\frac{1}{3^n}\right)-\left(2-\frac{1}{3^{n-1}}\right)=\frac{1}{3^{n-1}}-\frac{1}{3^n}=\frac{3}{3^n}-\frac{1}{3^n}=\frac{2}{3^n}

Note that the series turns out to be geometric, since

a_n=2\,(1/3)^n=a\,r^n.

Sums of infinite series[edit]

Find the value to which each of the following series converges.

  • \sum_{n=0}^{\infty} \frac{3}{4^n}
Show hint

sum of an infinite geometric series

Show answer only

4

Show full solution

The series is

\sum_{n=0}^{\infty} 3\left(\frac{1}{4}\right)^n

and so is geometric with first term a=3 and common ratio r=1/4. So

s=\frac{a}{1-r}=\frac{3}{1-1/4}=4.
  • \sum_{n=1}^{\infty} \left(\frac{2}{e}\right)^n
Show hint

sum of an infinite geometric series

Show answer only

\frac{2}{e-2}

Show full solution

s=\frac{2/e}{1-2/e}=\frac{2}{e-2}

  • \sum_{n=2}^{\infty} \frac{1}{n^2-n}
Show hint

telescoping series

Show answer only

1

Show full solution

Note that

\sum_{n=2}^{\infty} \frac{1}{n^2-n} = \sum_{n=2}^{\infty} \frac{1}{n(n-1)} = \sum_{n=2}^{\infty} \left(\frac{1}{n-1}-\frac{1}{n}\right)

by partial fractions. So

s = \lim_{N\to\infty} s_N = \lim_{N\to\infty} \left(1-\frac{1}{2}\right) + \left(\frac{1}{2}-\frac{1}{3}\right) + \left(\frac{1}{3}-\frac{1}{4}\right) + \ldots + \left(\frac{1}{N-1}-\frac{1}{N}\right).

All but the first and last terms cancel out, so

s = \lim_{N\to\infty} \left(1-\frac{1}{N}\right) = 1.
  • \sum_{n=1}^{\infty} \frac{(-1)^n 2^{n-1}}{3^n}
Show hint

rewrite so that all exponents are n

Show answer only

−1/5

Show full solution

The series simplifies to

\sum_{n=1}^{\infty} \frac{(-1)^n 2^n}{3^n \cdot 2} = \sum_{n=1}^{\infty} \frac{1}{2} \left(\frac{-2}{3}\right)^n

and so is geometric with common ratio r=-2/3 and first term -1/3. Thus

s=\frac{-1/3}{1-(-2/3)}=-1/5.

Convergence and divergence of infinite series[edit]

Determine whether each the following series converges or diverges. (Note: Each "Hint" gives the convergence/divergence test required to draw a conclusion.)

  • \sum_{n=1}^{\infty} \frac{1}{n^2}
Show hint

p-series

Show answer only

converges

Show full solution

This is a p-series with p=2. Since p>1, the series converges.

  • \sum_{n=0}^{\infty} \frac{1}{2^n}
Show hint

geometric series

Show answer only

converges

Show full solution

This is a geometric series with common ratio r=1/2, and so converges since |r|<1.

  • \sum_{n=1}^{\infty} \frac{n}{n^2+1}
Show hint

limit comparison test

Show answer only

diverges

Show full solution

This series can be compared to a p-series:

\sum_{n=1}^{\infty} \frac{n}{n^2+1} \sim \sum_{n=1}^{\infty} \frac{n}{n^2} = \sum_{n=1}^{\infty} \frac{1}{n}

The \sim symbol means the two series are "asymptotically equivalent"—that is, they either both converge or both diverge because their terms behave so similarly when summed as n gets very large. This can be shown by the limit comparison test:

\lim_{n\to\infty} \left( \frac{n}{n^2+1} \div \frac{1}{n} \right) = \lim_{n\to\infty} \left( \frac{n}{n^2+1} \cdot \frac{n}{1} \right) = \lim_{n\to\infty} \frac{n^2}{n^2+1} = 1

Since the limit is positive and finite, the two series either both converge or both diverge. The simpler series diverges because it is a p-series with p=1 (harmonic series), and so the original series diverges by the limit comparison test.

  • \sum_{n=2}^{\infty} \frac{1}{\ln n}
Show hint

direct comparison test

Show answer only

diverges

Show full solution

This series can be compared to a smaller p-series:

\sum_{n=2}^{\infty} \frac{1}{\ln n} \ge \sum_{n=2}^{\infty} \frac{1}{n}

The p-series diverges since p=1 (harmonic series), so the larger series diverges by the appropriate direct comparison test.

  • \sum_{n=0}^{\infty} \frac{n!}{2^n}
Show hint

divergence test

Show answer only

diverges

Show full solution

The terms of this series do not have a limit of zero. Note that when n>1,

\frac{n!}{2^n} = \frac{n}{2}\cdot\left[\frac{n-1}{2}\cdot\frac{n-2}{2}\dots\frac{2}{2}\right]\cdot\frac{1}{2} \ge \frac{n}{2}\cdot(1)\cdot\frac{1}{2} = \frac{n}{4}

To see why the inequality holds, consider that when n=2 none of the fractions in the square brackets above are actually there; when n=3 only 2/2 (which is the same as [n-1]/2) is in the brackets; when n=4 only 3/2 (equal to [n-1]/2) and 2/2 (equal to [n-2]/2) are there; when n=5, only 4/2, 3/2, and 2/2 are there; and so forth. Clearly none of these fractions are less than 1 and they never will be, no matter what n>1 is used.

The fact that

\lim_{n\to\infty} \frac{n}{4} = \infty

then implies that

\lim_{n\to\infty} \frac{n!}{2^n} = \infty

Therefore the series diverges by the divergence test.

  • \sum_{n=1}^{\infty} \frac{\cos\pi n}{n}
Show hint

alternating series test

Show answer only

converges

Show full solution

This is an alternating series:

\sum_{n=1}^{\infty} \frac{\cos\pi n}{n} = \sum_{n=1}^{\infty} \frac{(-1)^n}{n}

Since the sequence

|a_n|=\frac{1}{n}

decreases to 0, the series converges by the alternating series test.

  • \sum_{n=2}^{\infty} \frac{(-1)^n}{n\ln n-1}
Show hint

alternating series test

Show answer only

converges

Show full solution

Since the terms alternate, consider the sequence

|a_n|=\frac{1}{n\ln n-1}

This sequence is clearly decreasing (since both n and \ln n are increasing — one may also show that the derivative (with respect to n) of the expression is negative for n\ge2) and has limit zero (the denominator goes to infinity), so the series converges by the alternating series test.

Absolute and conditional convergence[edit]

Determine whether each the following series converges conditionally, converges absolutely, or diverges. (Note: Each "Hint" gives the test or tests that most easily lead to the final conclusion.)

  • \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}
Show hint

alternating series test and either direct comparison test or integral test

Show answer only

converges conditionally

Show full solution

This series alternates, so consider the sequence

|a_n|=\frac{1}{\sqrt{n}}

Since this sequence is clearly decreasing to zero, the original series is convergent by the alternating series test. Now, consider the series formed by taking the absolute value of the terms of the original series:

\sum |a_n|=\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}

This new series can be compared to a p-series:

\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \ge \sum_{n=1}^{\infty} \frac{1}{n}

Since the smaller series diverges, the larger one diverges. But this means the original (alternating) series was not absolutely convergent. (This last fact can also be shown using an integral test.) Therefore, the original series is only conditionally convergent.

  • \sum_{n=2}^{\infty} \frac{(-1)^n \ln n}{n}
Show hint

alternating series test and either integral test or direct comparison test

Show answer only

converges conditionally

Show full solution

This series alternates, so consider the sequence

|a_n|=\frac{\ln n}{n}

This sequence has a limit of zero by, for example, L'Hospital's Rule.

\lim_{n\to\infty}\frac{\ln n}{n}=\lim_{n\to\infty}\frac{1/n}{1}=0

That the sequence is decreasing can be verified by, for example, showing that as a continuous function of x, its derivative is negative.

\frac{d}{dx}\,\frac{\ln x}{x} = \frac{1 - \ln x}{x^2} < 0 \mbox{ if } x > e

This means that the terms definitely decrease starting with the second term (n=3). Thus, the series starting at n=3 is convergent by the alternating series test; clearly, then, the series starting at n=2 also converges (since the two series only differ by one term). Now, consider the series formed by taking the absolute value of the terms of the original series:

\sum |a_n|=\sum_{n=2}^{\infty} \frac{\ln n}{n}

This new series of positive terms only can be compared to a p-series:

\sum_{n=2}^{\infty} \frac{\ln n}{n} \ge \sum_{n=2}^{\infty} \frac{1}{n}

Since the smaller series diverges, the larger one diverges. Alternatively, the integral test can be used to test the convergence of the series of positive terms, since f(x) = \frac{\ln x}{x} is clearly a continuous, positive function on [2,\infty) and, as we have just verified, is also decreasing:

\int_{2}^{\infty} \frac{\ln x}{x}\,dx = \lim_{t\to\infty} \int_{2}^{t} \frac{\ln x}{x}\,dx = \lim_{t\to\infty} \int_{\ln 2}^{\ln t} u\,du

by the substitution u=\ln x; and this last expression becomes

\lim_{t\to\infty} \Bigl(\frac{1}{2} u^2\bigr|_{\ln 2}^{\ln t}\Bigr) = \lim_{t\to\infty} \frac{1}{2}(\ln t)^2 - \frac{1}{2} (\ln 2)^2 = \infty

Since the improper integral diverges, the series of positive terms diverges.

Either way you test it, the series with all positive terms diverges, and this means the original (alternating) series was not absolutely convergent. Thus, the original series is only conditionally convergent.

  • \sum_{n=2}^{\infty} \frac{(-1)^n n}{(\ln n)^2}
Show hint

divergence test

Show answer only

diverges

Show full solution

This series is alternating, but note that by L'Hospital's Rule

\lim_{n\to\infty}\frac{n}{(\ln n)^2} = \lim_{n\to\infty}\frac{1}{2(\ln n)(1/n)} = \lim_{n\to\infty}\frac{n}{2\ln n}
\mbox{ } = \lim_{n\to\infty}\frac{1}{2/n} = \lim_{n\to\infty}\frac{n}{2} = \infty

Which implies that

\lim_{n\to\infty}\frac{(-1)^n n}{(\ln n)^2}

does not exist, and hence by the divergence test, the series diverges.

  • \sum_{n=1}^{\infty} \frac{(-1)^n 2^n}{e^n-1}
Show hint

limit comparison test with geometric series

Show answer only

converges absolutely

Show full solution

While this alternating series can be shown to converge by the alternating series test, it can also be shown that the absolute value of the terms form a convergent series, and this is sufficient to conclude absolute convergence of the original series. Thus we will skip the former test and show only the latter.

\sum_{n=1}^{\infty} |a_n| = \sum_{n=1}^{\infty} \frac{2^n}{e^n-1}

This series of positive terms is asymptotically geometric with r=2/e:

\sum_{n=1}^{\infty} \frac{2^n}{e^n-1} \sim \sum_{n=1}^{\infty} (2/e)^n

The equivalence of these series is shown by using a limit comparison test:

\lim_{n\to\infty} \left[\frac{2^n}{e^n-1} \div (2/e)^n\right] = \lim_{n\to\infty} \left(\frac{2^n}{2^n}\cdot\frac{e^n}{e^n-1}\right) = \lim_{n\to\infty} \frac{1}{1-1/e^n} = 1

Since the limit is positive and finite, and since the simpler series converges because it geometric with r=2/e (the absolute value of which is less than 1), then the series of positive terms converges by the limit comparison test. Thus the original alternating series is absolutely convergent.

Note, by the way, that a direct comparison test in this case is more difficult (although still possible to do), since

\sum_{n=1}^{\infty} \frac{2^n}{e^n-1} \ge \sum_{n=1}^{\infty} (2/e)^n

and we need the inequality to go the other way to get a conclusion, since the geometric series converges.

This can be fixed by choosing the new series more carefully:

\sum_{n=1}^{\infty} \frac{2^n}{e^n-1} \le \sum_{n=1}^{\infty} \frac{2^n}{e^n-.5e^n} = \sum_{n=1}^{\infty} \frac{2^n}{.5e^n} = \sum_{n=1}^{\infty} 2(2/e)^n

Comparing the original series to the new (convergent geometric) series gives the desired result.

  • \sum_{n=1}^{\infty} \frac{(-1)^n}{\sin^2 n}
Show hint

divergence test

Show answer only

diverges

Show full solution

Since this is an alternating series, we can try the alternating series test. Consider the absolute value of the terms:

|a_n|=\frac{1}{\sin^2 n}

Because

\lim_{n\to\infty} \sin^2 n

does not exist (since it continually oscillates within the interval [0, 1] as n gets larger)

\lim_{n\to\infty} \frac{1}{\sin^2 n}

doesn't exist either. Thus the alternating series test fails (it is inconclusive).

However, in such a situation we can use the divergence test instead. Since

\lim_{n\to\infty} \frac{(-1)^n}{\sin^2 n}

also does not exist (and thus the terms of the series do converge to 0), the original series diverges by the divergence test.

  • \sum_{n=1}^{\infty} \frac{(-1)^n n!}{(2n)!}
Show hint

ratio test

Show answer only

converges absolutely

Show full solution

Because of the factorials in this series, we try the ratio test:

\lim_{n\to\infty}\left|\frac{(-1)^{n+1}(n+1)!}{[2(n+1)]!}\div\frac{(-1)^n n!}{(2n)!}\right| = \lim_{n\to\infty}\frac{(n+1)!}{n!}\cdot\frac{(2n)!}{(2n+2)!} = \lim_{n\to\infty}\frac{(n+1)}{(2n+2)(2n+1)} = \lim_{n\to\infty}\frac{1}{2(2n+1)} = 0

Since the limit is less than 1, the series converges absolutely by the ratio test.

  • \sum_{n=1}^{\infty} \frac{(-1)^n e^{1/n}}{\arctan n}
Show hint

divergence test

Show answer only

diverges

Show full solution

Although this is an alternating series, neither the numerator nor the denominator have infinite limits, so it is likely that a divergence test will work.

Note that

\lim_{n\to\infty} \frac{e^{1/n}}{\arctan n} = \frac{e^0}{\pi/2} = \frac{2}{\pi} \ne 0

Thus

\lim_{n\to\infty} \frac{(-1)^n e^{1/n}}{\arctan n} \ne 0

In fact, the latter limit does not exist. So, by the divergenve test, the series diverges.