# Real Analysis/Section 2 Exercises

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- For the following list of sequences determine whether the following sequences converge or diverge directly from the definition of convergence:
- the sequence ;
- the sequence ;
- the sequence ;
- The recursive sequence defined by the following:

- Let
- .

*x*_{n}. - For the following sequences determine for which real values
*x*does the given sequence converge, and what the sequence converges to:- ;
- ;
- for an arbitrary real number
*x*;

- Given any real number
*c*, find a recursively defined sequence that converges to . - Given a sequence (
*x*_{n}), and a natural number*k*, define a sequence*y*_{n}by*y*_{n}=*x*_{n+k}. Show that (*x*_{n}) is convergent if and only if (*y*_{n}) is convergent. Show further that when they converge they converge to the same limit. - Suppose that the sequences (
*x*_{n}) and (*y*_{n}) converge to a real number*a*. Show that the sequence (z_{n}) defined by*z*_{n}→*a*. - Let (
*x*_{n}) be a sequence of real numbers and let (*y*_{n}) be a sub-sequence. Suppose (*y*_{n}) is convergent, show that (*x*_{n}) may not necessarily be convergent. - Suppose that (
*x*_{n}) is a convergent sequence that does not converge to 0. Further assume that for all*n*in**N**,*x*_{n}≠ 0. Show that there exists δ > 0 so that |*x*_{n}| > δ and |lim*x*_{n}| > δ. - Cesaro Mean convergence: We say a sequence (
*x*_{n}) converges to*x***by Cesaro means**if the sequence of averages*y*_{n}= (*x*_{1}+*x*_{2}+ … +*x*_{n})/*n*converges to*x*. Suppose (*x*_{n}) converges to a real number*x*, show that (*x*_{n}) converges by Cesaro means to*x*. Give an example to show that a divergent sequence (*x*_{n}) may converge by Cesaro means. - Find the sequence of Cesaro means for (1, 1, -1, 1, 1, -1...) and determine if the converge. If they converge, find the limit.
- Consider the recursively defined sequence given by
*x*_{1}= 1 and*x*_{n}= 1 + 1/*x*_{n}. Show that*x*_{n}converges and find its limit. - In our discussion of telescoping series we showed that a telescoping series converged to
*a*_{1}− lim*a*_{N+1}and for this to hold it was not necessary to have that lim*a*_{N+1}= 0. Indeed, it is correct that this is not necessary. On the other hand we later proved that for a convergent series the limit of the terms must be 0. How can both by correct? Explain why, in our set up, we may have a convergent telescoping series such that lim*a*_{N+1}≠ 0, but it is still true that for every convergent series the limit of the terms is 0. - Suppose that
*c*_{n}≤*a*_{n}≤*b*_{n}for all natural numbers*n*. Show that if both ∑*c*_{n}and ∑*b*_{n}converge, then ∑*a*_{n}converges.