It?s also called the Remainder Estimation of Alternating Series.

This is to calculating (approximating) an Infinite Alternating Series:

?Jump over to Khan academy for practice: Alternating series remainder

?Refer to The Organic Chemistry Tutor: Alternate Series Estimation Theorem?Refer to Mathonline: Error Estimation for Approximating Alternating Series?Refer to mathwords: Alternating Series Remainder

The logic is:

- First to test the series? convergence.
- If the series CONVERGES, then we can proceed to calculate it by Error Estimation Theorem. Otherwise we aren?t able to.
- We can express the series as the sum of partial sums & infinite remainder:

(? Sn is the first n terms, and Rn is from the n+1 term to the rest terms.)

- And the ?structure? in the partial sum & remainder is:

- With a little twist, we will get the whole idea:

(?Since the Rn is the gap between S & Sn, so we call it The Error)

- ? And the theorem is: The Remainder MUST NOT be greater than its first term:

?Actual sum = Partial sum + Remainder: refer to Khan academy: Alternating series remainder

## Sign & Size of Error

?Refer to Khan academy: Alternating series remainder?Refer to Khan academy: Worked example: alternating series remainder

For the Remainder series, its FIRST TERM is always DOMINATING the whole remainder:

- It dominates the remainder?s SIGN: positive or negative.
- It dominates the remainder?s SIZE: the whole remainder?s absolute value CAN?T BE greater than the first term.

Based on the error?s sign, we could tell the approximated series is UNDERESTIMATED or OVERESTIMATED:

- If Error > 0, then the approximated series is Underestimate.
- If Error < 0, then the approximated series is Overestimate.

## Bound the Error (accuracy control)

The error bound regards to the accuracy of the approximated series, and we want to control the accuracy before approximation.

?Refer to Khan academy: Worked example: alternating series remainder

We have 2 ways to bound the error in a range:

- Set up how small we want the error to be, or
- Set up how many terms we want to have in the partial sum Sn.

## Bound by terms

The Larger n ? The smaller gap ? The lesser Error ? The more accurate.

Strategy:

- We could set the partial sum to include a certain number of terms, etc. 100 terms
- And the first term of Remainder should be the 101st term.
- The error bound, or the error is dominated by the first term.
- So we say the error bound IS the value of the 101st term.

## Bound the error

To bound the error in a range, we often say:

- ?Approximate the series to the 2 decimal places?,
- ?Let the error be less than 0.01?,
- ?We want the accuracy within 0.01?

What they mean are the same:

? And by solving the inequality, we will get the scope for n, then get the Smallest Integer of n in that scope.

## Example

Solve:

- First to notice, the partial sum is already set to 100 terms, so we’re to control accuracy by bound the terms.
- So the error should be from the 101st term to infinity.
- But the error bound is actually dominated by the first term of the error.
- So the error bound = the value of 101st term:

- We could say that: The error bound is negative, and negative error causes overestimation.

## Example

Solve:

- It?s clear this is a alternating series.
- So we want to do the alternating series test first, and it passed, which means it converges.
- Since the series converges, we can do further approximation.
- See that we don?t know how many terms are in the partial sum, and only know how much accurate we’d like.
- So we?re to approximate by bound the error, and find out the terms.
- Apply the Error Approximation Theorem, assume the first term of remainder is a_(n+1):

- Solve out the inequality to get n ? 999,999
- And 999,999 the smallest integer of n to make the series converges with 2 decimal accuracy.