Error Estimates for Alternating Term Series

In a Nut Shell: The partial sum, S_n. of any convergent series can be used to approximate

the total sum of the series. The conditions for an alternating series of the form

∑ (˗1) ^{n ˗1} a_n

to converge are: 0 ≤ a_n+1 ≤ a_n and lim a_n = 0

n→∞

then the remainder R_n is s ˗ s_n and

| R_n | = | s ˗ s_n | ≤ a_n+1

Note: The error, | Rn |, is less or equal to the first term neglected, namely a_n+1 .

This result | R_n | = | s ˗ s_n | ≤ a_n+1 is referred to as the Alternating Series

Estimation Theorem.

Example: Find the number of terms needed to find the sum of

∞

∑ (˗1 ⁿ / ( 3ⁿ n! ) to four decimal places.

n = 1

Note that the alternating series converges since for a_n = 1/(3ⁿ n!) satisfies both condition

of the alternating series test.

a₆ = 1/(3⁶ 6!) ~ 0.0000019

s₅ = ˗ 1/3 + 1/(3² 2!) ˗ 1/(3³ 3!) + 1/3⁴ 4!0 ˗1/3⁵ 5!) = ˗ 0.283471

So adding a₆ does not change value of s. So sum = ˗ 0.2835 to four decimals

and you need 5 terms (result)

Example: Find the number of terms needed to find the sum of

∞

∑ (˗1⁾ⁿ/(5ⁿ n) | error | < 0.0001

n = 1

Note that the alternating series converges since for a_n = 1/(5ⁿ n) both conditions

are satisfied in the alternating series test.

Now a₅ = ˗ 1/(5⁵ 5) = ˗ 0.000064

s₄ = ˗ 1/5 + 1/(5² 2) ˗ 1/(5³ 3) + 1/(5⁴ 4) = ˗ 0.1822666

The error is less than | ˗ 0.000064 | < 0.0001 So only need 4 terms. (result)

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