Chapter 12: Q 2. (page 824)

If $|r| < 1$ , the sum of the geometric series a
$\sum_{k = 1}^{\infty} a r^{k - 1}$ is ________ .

Short Answer

Expert verified

If $|r| < 1$ , the sum of the geometric series a $\sum_{k = 1}^{\infty} a r^{k - 1}$ is $\frac{a_{1}}{(1 - r)}$ .

Step by step solution

Step 1. Given information.

If $|r| < 1$ , the sum of the geometric series a $\sum_{k = 1}^{\infty} a r^{k - 1}$ is _________.

A geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Step 2. Fill in the blanks.

The sum $s_{n}$ of first $n$ terms of a geometric sequence is given by

$s_{n} = \frac{a_{1} (1 - r^{n})}{(1 - r)} s_{n} = \frac{a_{1}}{(1 - r)} - \frac{a_{1} r^{n})}{(1 - r)}$

If $|r| < 1$ then $|r^{n}|$ approaches $0$ as $n \to \infty$ then the term $\frac{a_{1} r^{n}}{(1 - r)}$ approaches $0$ and $s_{n}$ approaches $\frac{a_{1}}{(1 - r)}$ as $n \to \infty$ .

Therefore, if $|r| < 1$ the sum of the geometric series a $\sum_{k = 1}^{\infty} a r^{k - 1}$ is $\frac{a_{1}}{(1 - r)}$ .

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If $|r| < 1$ , the sum of the geometric series a
$\sum_{k = 1}^{\infty} a r^{k - 1}$ is ________ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Fill in the blanks.

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If r<1, the sum of the geometric series a∑k=1∞ark-1is ________ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Fill in the blanks.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

If $|r| < 1$ , the sum of the geometric series a
$\sum_{k = 1}^{\infty} a r^{k - 1}$ is ________ .