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Cesaro Summability Given an infinite series Ean let Sn be the sequence of partial sums and letNote that this is the correct definition. series is Cesaro summable i...

Question

Cesaro Summability Given an infinite series Ean let Sn be the sequence of partial sums and letNote that this is the correct definition. series is Cesaro summable if limn_x Gn exists (and is finite) and this limit is called the Cesaro sum_ Given the series Ca find the Cesaro sum for |-l <1.

Cesaro Summability Given an infinite series Ean let Sn be the sequence of partial sums and let Note that this is the correct definition. series is Cesaro summable if limn_x Gn exists (and is finite) and this limit is called the Cesaro sum_ Given the series Ca find the Cesaro sum for |-l <1.



Answers

Show that if $\sum a_{n}$ converges, then $\lim _{n \rightarrow \infty} a_{n}=0$. [Hint: Consider $\lim _{n \rightarrow \infty}\left(S_{n}-S_{n-1}\right),$ where $S_{n}$ is the $n^{\text {th }}$ partial sum.

Hello. Let's let? S even be a secret of partial sums for the convergent series. We have this series and going from one to infinity of estaban is equal to L. Then we get the limit as N approaches infinity of estaban well is equal to L. So since we have that are suburban is equal to the sum where we have K which is equal to n plus one to infinity. Um uh A sub K we get to our CNN is equal to l minus S M N. So then we get that the limit as N approaches infinity of our suburban. Well, it's equal to the limit as N approaches infinity of l minus as cement. So then we get that are seven is equal to the limit as um is equal to the limit as N approaches infinity of l minus. The limit as an approaches infinity of S M N. Which we get then um the limit as an approaches infinity of our seven is equal to well l minus L which is equal to zero. So limit as N approaches infinity of our cement, We then get is equal to zero

We want to find the end partial some and evaluate experiment to determine whether or not the given series converges the series in question is the sum from N equals one to infinity of negative to raise to the power of that. Remember the end partial sum is simply pound by determining how the some transitions between each end. So we want to take from N equals one or rather from I equals one up to some end what our series value is. So the impartial some Sn is this something I equals one to end of negative to the I We can rewrite this as two thirds times negative, two to the end minus one. Now that we've written there and partial some in the form of expression related to end rather than interest related to the summer. I we can take the limit and determine the convergence so the limit as N approaches infinity. Sn eliminate N approaches infinity of two thirds negative to the n minus one. This is just negative to the infinity or plus or minus infinity. Since the limit diverges, that means the series also diverted


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