# Error in Series Approximation Proof

• Potatochip911
In summary: If ##a_{n+1}>0##, how do ##a_{n+1}## and ##a_{n+1} + a_{n + 2}## compare? By that, I mean, can you determine which expression is larger?It might help your intuition to think about a specific alternating series, such as 1 - 1/2 + 1/3 - 1/4 + 1/5 -+ ..., and grouping terms both ways as suggested in the hint.Yes, you can compare the magnitude of the terms by taking the absolute value.
Potatochip911

## Homework Statement

Prove:

Hint: Group the terms in the error as ##(a_{n+1}+a_{n+2})+(a_{n+3}+a_{n+4})+\cdots## to show that the error has the same sign as ##a_{n+1}##. Then group them as ##a_{n+1}+(a_{n+2}+a_{n+3})+(a_{n+4}+a_{n+5})+\cdots## to show that the error has magnitude less than ##\left| a_{n+1} \right |##.

## Homework Equations

3. The Attempt at a Solution [/B]
From the first part of the hint I can see that the grouped terms ##(a_{n+1}+a_{n+2})+\cdots## will take the sign of the first term because ##\lim_{n\to\infty}a_{n}=0## therefore, ##\left | \lim_{n\to\infty}a_{n+1} \right | > \left | \lim_{n\to\infty}a_{n+2} \right | ## At this point I am not really sure what to do with the second part of the hint although I'm assuming it has something to do with the less than or equal to symbol.

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Potatochip911 said:

## Homework Statement

Prove:

Hint: Group the terms in the error as ##(a_{n+1}+a_{n+2})+(a_{n+3}+a_{n+4})+\cdots## to show that the error has the same sign as ##a_{n+1}##. Then group them as ##a_{n+1}+(a_{n+2}+a_{n+3})+(a_{n+4}+a_{n+5})+\cdots## to show that the error has magnitude less than ##\left| a_{n+1} \right |##.

## Homework Equations

3. The Attempt at a Solution [/B]
From the first part of the hint I can see that the grouped terms ##(a_{n+1}+a_{n+2})+\cdots## will take the sign of the first term because ##\lim_{n\to\infty}a_{n}=0##
No, that's not it. You are not using the fact that this is an alternating series, where |an + 1| < |an|.
Potatochip911 said:
therefore, ##\left | \lim_{n\to\infty}a_{n+1} \right | > \left | \lim_{n\to\infty}a_{n+2} \right | ## At this point I am not really sure what to do with the second part of the hint although I'm assuming it has something to do with the less than or equal to symbol.

Mark44 said:
No, that's not it. You are not using the fact that this is an alternating series, where |an + 1| < |an|.
Edit: Confused myself
Edit #2: I don't understand I thought that's what I used to show that if ##a_{n+1}>0## then ##a_{n+1}+a_{n+2}>0##

Last edited:
Potatochip911 said:
Edit: Confused myself
Edit #2: I don't understand I thought that's what I used to show that if ##a_{n+1}>0## then ##a_{n+1}+a_{n+2}>0##
Meant to edit my last post...

Mark44 said:
No, that's not it. You are not using the fact that this is an alternating series, where |an + 1| < |an|.

Potatochip911 said:
Edit #2: I don't understand I thought that's what I used to show that if ##a_{n+1}>0## then ##a_{n+1}+a_{n+2}>0##
I don't see how this helps your cause.

If ##a_{n+1}>0##, how do ##a_{n+1}## and ##a_{n+1} + a_{n + 2}## compare? By that, I mean, can you determine which expression is larger?

It might help your intuition to think about a specific alternating series, such as 1 - 1/2 + 1/3 - 1/4 + 1/5 -+ ..., and grouping terms both ways as suggested in the hint.

Mark44 said:
I don't see how this helps your cause.

If ##a_{n+1}>0##, how do ##a_{n+1}## and ##a_{n+1} + a_{n + 2}## compare? By that, I mean, can you determine which expression is larger?

It might help your intuition to think about a specific alternating series, such as 1 - 1/2 + 1/3 - 1/4 + 1/5 -+ ..., and grouping terms both ways as suggested in the hint.
Well I thought I had shown that the magnitude of ##a_{n}## was larger than ##a_{n+1}## by assuming than ##a_{n}## is positive and then stating that ##a_{n}+a_{n+1}>0##.

I have that if the last term I include in my calculation is ##a_{n}## and ##a_{n}## is negative, and also using the fact that ##a_{1}>a_{2}>\cdots>a_{n}>a_{n+1}>a_{n+2}>\cdots>a_{k}##
Then for the remainder, i.e. the terms omitted from the calculation, ##R_{n}=a_{n+1}+a_{n+2}+a_{n+3}+a_{n+4}+\cdots+a_{k}##

Grouping the terms like ##R_{n}=(a_{n+1}+a_{n+2})+(a_{n+3}+a_{n+4})+\cdots+(a_{k-1}+a_{k})## it can be seen that the remainder has to be a positive number since the first term in the parenthesis will always be positive and larger than the 2nd term in the parenthesis which is negative.

Next by regrouping the terms like ##R_{n}=a_{n+1}+(a_{n+2}+a_{n+3})+(a_{n+4}+a_{n+5})+\cdots## it can be seen that after the first term ##a_{n+1}## all of the parenthesis will be negative therefore the remainder will be less than ##a_{n+1}##. So I have that ##R_{n}## is greater than 0 and less than ##a_{n+1}## if the last term included in my calculation is negative.
##S=S_{n}+R_{n} \Longrightarrow S-S_{n}=R_{n}## now since ##0<R_{n}<a_{n+1}## I can replace ##R_{n}## with ##a_{n+1}## and change the sign to an inequality which gives $$S-S_{n}<a_{n+1}$$ At this point I was wondering if I could just take the absolute value of both sides to finish the proof or do I have to go over the other case where the last term I calculate is positive.

## What is an error in series approximation?

An error in series approximation is the difference between the exact value and the approximate value obtained by using a finite number of terms in a series expansion.

## How is error in series approximation calculated?

The error in series approximation is calculated using the remainder term, which is the difference between the exact value and the value obtained by truncating the series at a certain term.

## What is the purpose of error in series approximation?

The purpose of error in series approximation is to determine the accuracy of an approximation and to understand how many terms are needed in a series expansion to obtain a desired level of accuracy.

## What factors can influence the error in series approximation?

The error in series approximation can be influenced by the number of terms used in the series expansion, the magnitude of the terms in the series, and the behavior of the function being approximated near the point of expansion.

## How can the error in series approximation be reduced?

The error in series approximation can be reduced by increasing the number of terms used in the series expansion, using a more accurate series expansion, or choosing a different point of expansion. Additionally, using techniques such as Taylor's theorem or estimating the error can also help reduce the error in series approximation.

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