1. Let t be a conditionally convergent series, that is, t x„ converges, but E Ixn1 does not. a. Prove that E fnl.. 0, there exist infinitely many positive terms and negative terms of (x„) with...

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1. Let t be a conditionally convergent series, that is, t x„ converges,
but E Ixn1 does not. a. Prove that E fnl.. 0, there exist infinitely many positive terms and negative terms of (x„) with absolute valve less than e. c. Using parts (a) and (b), prove that for any conditionally convergent series E x„, and any real member c, there is a rearrangement of (x„)
such that E
2. For each m E N, let f„,(x) = nliMCOS(171!7/X)2n. Does this sequence of functions (f„,(x))„,,,, converge pointwise? If so, what is its limit? Does it converge uniformly?
3. Let f be defined for all real x and suppose that for each x,y E R, 11(o) - (0 - . Prove that f is constant.

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Answered Same DayDec 22, 2021

Answer To: 1. Let t be a conditionally convergent series, that is, t x„ converges, but E Ixn1 does not. a....

Robert answered on Dec 22 2021

121 Votes

Solutions
1.(a) Note that ∑
n
xn =
∑
{n | xn>0}
xn +
∑
{n | xn<0}
xn
Lets call the positive part as
∑
x+n and negative part as
∑
x−n . This can be easily defined to be:
x+n =
{
xn xn > 0,
0 xn < 0
For the convergence of
∑
x+n and
∑
xn− four possibilities exist:
•
∑
x+n converges and
∑
x−n converges.
•
∑
x+n converges and
∑
x−n diverges.
•
∑
x+n diverges and
∑
x−n converges.
•
∑
x+n diverges and
∑
x−n diverges.
We cant have case 1. Because if we had then
∑
xn would be an absolutely convergenet series, not conditionally
convergent. We can’t have case 2 as well, because if we add both the convergent series
∑
x+n and the divergent
series
∑
x−n , the resulting series
∑
xn will diverge. But...

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1. Let t be a conditionally convergent series, that is, t x„ converges, but E Ixn1 does not. a. Prove that E fnl.. 0, there exist infinitely many positive terms and negative terms of (x„) with...

Answer To: 1. Let t be a conditionally convergent series, that is, t x„ converges, but E Ixn1 does not. a....

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