Error Estimates for Positive Term Series

In a Nut Shell: The remainder, R_n , of the integral test can be used to find the number of

terms in a positive term series needed to approximate the sum of the series to within a

specified tolerance provided that the positive terms series satisfies conditions of the

integral test. Click here for a reminder of conditions needed to apply the integral test.

The estimate of the remainder, R_n, is as follows:

∞ ∞

∫ f(x) dx ≤ R_n ≤ ʃ f(x) dx

n + 1 n

Use the remainder, R_n, to determine the number of terms needed to approximate the sum

of a series to within a specified tolerance.

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

∞

∑ 10/n⁸ to three decimals.

n = 1

Note that the series is a positive term p-series with p = 8. Therefore the series converges

and has a sum. Also the series satisfies the conditions for the integral test. Note the

improper integral expression for R_n .

∞ t

R_n ≤ ʃ (10/x⁸) dx = lim (10 x^˗7 / ˗7) | = (10/7) n^˗7 < 0.0005

x = n t→∞ n

Note: To estimate the sum to within three decimal places, one must make certain

the remainder is less than 0.0005.

Solve for n. n⁷ > 2857.15 n > 3.1 So n = 4

R₄ = 10/1⁸ + 10/(2⁸) + 10/(3⁸) + 10/(4⁸) ~ 10.041 (result)

after rounding to three decimal places

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