Chapter 10.1, Problem 69E

Formulas for sequences of partial sums Consider the following infinite series.

1. a. Find the first four partial sums S₁, S₂, S₃, and S₄ of the series.
2. b. Find a formula for the nth partial sum S_n of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, and S₈ of the infinite series.
3. c. Make a conjecture for the value of the series.

69. ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}90{\left(0.1\right)}^k}$