# Solving Convergent Series Problem: Proving Limit of nan is 0

• steven187
In summary, the author has been working on a problem where the sequence Nan goes to zero. They showed that the limit of the sequence is 0, and then proved the same result for even numbers. Additionally, by assuming decreasingness, it also follows for odd numbers.
steven187
hello all

iv been workin on this problem its kind of awkward check it out

{an} is a decreasing sequance, an>=0 and there is a convergent series Sn with terms an
we need to prove that the limit of nan is 0

i first started of a sequence bn=an+1+an+2+...+a2n
then I showed that the limit of bn is also 0 where do i go from here

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use the hypothesis, that they are decreasing. this immediately implies your result for even n, and then also the result for all n.

[how did you think of that way of beginning? was it a hint? i.e. the first step was the clever part.]

well in terms of how i began i was trying to think of a sequence that converges to zero so that i can bound the sequence nan from below and i was trying to look for something else to bound it by from above so that i can use either the comparision test or the squeze theorem but no matter how much i play with it i can't find anything to bound it by from above so far this is what iv done

bn=an+1+an+2+...+a2n then i showed
lim n->infinity bn= 0
bn=an+1+an+2+...+a2n
<=an+an+...+an
=nan this is where i get confused I obviously can't bound it by the sequence an because nan>=an so where do i go from here

by the way how do you use this latex stuff I am sure it would be pretty useful and quik

what you have shown, proves the result for even n, assuming decreasingness. then it also follows for odd n, again using decreasingness.

i.e. 2n a(n) goes to zero if and only if n a(n) does.

## 1. What is a convergent series?

A convergent series is a mathematical series in which the terms of the series approach a finite limit as the number of terms increases.

## 2. How is the limit of a convergent series determined?

The limit of a convergent series is determined by evaluating the sum of the terms of the series as the number of terms approaches infinity. If the sum approaches a finite number, then the limit is considered to be that number.

## 3. How is the limit of nan (not a number) determined for a convergent series?

In order to determine the limit of nan for a convergent series, we must first understand what nan represents. Nan stands for "not a number" and it is typically used to represent mathematical expressions that are undefined or do not have a numerical value. In the case of a convergent series, if the limit of the terms of the series is undefined or does not have a numerical value, then the limit of nan is considered to be 0.

## 4. What is the significance of proving the limit of nan is 0 for a convergent series?

Proving that the limit of nan is 0 for a convergent series is important because it confirms that the series is converging to a finite value. This allows us to make accurate predictions and calculations based on the series.

## 5. What are some common techniques used to prove the limit of nan is 0 for a convergent series?

One common technique used to prove the limit of nan is 0 for a convergent series is the comparison test, which involves comparing the given series to a known convergent series. Other techniques include the ratio test, the root test, and the integral test.

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