Q. 64 Prove that every convergent geom... [FREE SOLUTION]

Chapter 7: Q. 64 (page 654)

Prove that every convergent geometric series converges absolutely.

Short Answer

Expert verified

To prove, first we need to consider a geometric series and check if its absolutely convergent when when applying limits.

Step by step solution

Step 1. Given information.

Consider the given question,

Every convergent geometric series converges absolutely.

Step 2. Consider the convergent series.

Consider the convergent geometric series ${鈭�}_{k = 1}^{鈭�} c r^{k}$ .

The series ${鈭�}_{k = 1}^{鈭�} c r^{k}$ is absolutely convergent when $\lim_{k 鈫� 鈭�} |\frac{a_{k + 1}}{a_{k}}| < 1$ .

The geometric series data-custom-editor="chemistry" ${鈭�}_{k = 1}^{鈭�} c r^{k}$ is convergent. Then, $|r| < 1$ .

The ratio is given by,

$|\frac{a_{k + 1}}{a_{k}}| = |\frac{c r^{k + 1}}{c r^{k}}| = |r|$

Step 3. Find the value of the limit.

The value of $\lim_{k 鈫� 鈭�} |\frac{a_{k + 1}}{a_{k}}|$ is given below,

$\lim_{k 鈫� 鈭�} |\frac{a_{k + 1}}{a_{k}}| = \lim_{k 鈫� 鈭�} |r| = |r|$

Which is greater than $1$ .

Thus, $\lim_{k 鈫� 鈭�} |\frac{a_{k + 1}}{a_{k}}| < 1$ .

Therefore, the series ${鈭�}_{k = 1}^{鈭�} c r^{k}$ converges absolutely.

Hence, every geometric convergent series is absolutely convergent.

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91影视

Prove that every convergent geometric series converges absolutely.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the convergent series.

Step 3. Find the value of the limit.

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