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


∑ (from n = 2 to ∞) -2/[n(n+1)]

∑ (from n=1 to ∞) (1 / √(n + 1)) diverges

b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?

Series with Geometric Terms

In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.

∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]

Find the sum of each series in Exercises 45–52.

∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]

Find the value of b for which

1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.

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.

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