Alternating Series

If the absolute value of the series converges

the series converges absolutely.

If the absolute value of the series does not converge

alternating series test

Step 1 of Alternating Series Test

take the limit of the absolute value.

Step 2 of the Alternating Series Test

find if the sequence of the absolute value decreases.

To determine if a series/sequence decrease

an+1 < an, cross multiply, and move terms to one side.

If Step 1 and Step 2 are correct

the series converges conditionally

If Step 1 and/or Step 2 are wrong

the series diverges.

Formula for Estimating Sums

|Sn - L| < | an + 1 |

To estimate a sum with an error range

1. write out terms until one | an| is less than 1/1000

2. Add up previous sums

If the absolute values of the series converges,

the alternating series also converges.

If the series is an alternating series and the absolute value series diverges

then then the alternating series cannot converge absolutely

If the series is an alternating series and the limit of | an | as n goes to infinity does not equal 0

then the series diverges.

If positive An and the series converges

the series converges absolutely