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

Make up an infinite series of nonzero terms whose sum is

b. −3

Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if

b. a = 13/2

Find the value of b for which

1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.

Convergent Series

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

∑ (from n = 2 to ∞) -2/[n(n+1)]

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.

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