b. From Example 5, Section 10.2, show that
S = 1 + ∑(from n=1 to ∞) [1 / (n²(n + 1))].

Convergent Series

Find the sums of the series in Exercises 19–24.

∑ (from n = 2 to ∞) -2/[n(n+1)]

Estimate the value of ∑ (from n=2 to ∞) (1 / (n² + 4)) to within 0.1 of its exact value.

In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]

Error Estimation

In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1

Compute the first four partial sums and find a formula for the $n^{\mathrm{th}}$ partial sum.

${\sum }_{n=1}^{\infty }2n-1$

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 1 to ∞)2ᵏ / 3ᵏ⁺²

Express 0.314141414… as a ratio of two integers.

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) √k / (ln¹⁰ k)