Ch 8.5 Alternating Series

2/14/24 & 2/15/24

Alternating series

series whose terms are alternately positive and negative

Let a_n>0. The alternating series \sum_{n=1}^{\infty }(-1)^na_n and \sum_{n=1}^{\infty }(-1)^{n+1}a_n converge when …

• \lim_{n \to \infty } a_n=0

• a_{n+1}\le a_n for all values of n

How can I prove if a_{n+1}\le a_n for all values of n?

Way 1

• let f(x) = inner part of series but not alternating series part

• Find f’(x)

• use f’(x) to prove if a_{n+1}\le a_n for all values of n

Way 2

• Cross multiply a_{n+1} & a_n

• simplify the terms

• whichever simplified term is biggest is also the biggest regular term

• the numerator used in the biggest simplified term is the numerator of the biggest term

What if \lim_{n \to \infty } a_n\neq0 ie the first condition fails

Use the n^{th} term test!

Remainder

Difference between the exact and approximate sum of the series.

R_n = S - S_n

aka error

If a series passes the alternating series test than the absolute value of the remainder is …

less than or equal to the first neglected term (ie first term not used in partial sum estimation)

\left| R_n \right|=\left| S-S_n \right|

\left| R_n \right|\le a_{n+1}

How to find a certain error

\left| R_n \right|\le a_{n+1}

R_n = the error

if the error needs to be less than a #, ex: c

a_{n+1}<c

Now isolate n

ex: # < n

so for an error less than c we need # +1 terms.

Absolute Convergence

If \sum_{}^{}\left| a_n \right| converges then \sum_{}^{}a_n also converges and the series is said to be absolutely convergent.

Conditionally Convergent

If \sum_{}^{}a_n converges but \sum_{}^{}\left| a_n \right| diverges the series is said to be conditionally convergent.