Repeating Decimals
Express each of the numbers in Exercises 23–30 as the ratio of two integers.
3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...

∑ (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?

Finding nth Partial Sums

In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.

2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …

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) ]

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