27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 1 to ∞)2ᵏ / 3ᵏ⁺²

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

b. From Example 5, Section 10.2, show that

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

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

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

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)

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) (((1/6)ᵏ + (1/3)ᵏ) × k⁻¹)

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

∑ (k = 0 to ∞) k / (2k + 1)