Chapter 7: Q. 10 (page 614)

Let $\sum_{k = 1}^{\infty} {Cr}^{k}$ be a series with c and $r$ $\in ℝ$ . Explain why the convergence of this series depends only upon the magnitude of r and not c .

Short Answer

Expert verified

The geometric behaviour is independent of c . The ratio is dependent on r . Therefore the convergence of the series is governed by the ratio r only .

Step by step solution

Step 1. Given information

We have been given a series to find out the parameters on which its convergence depends

Step 2. Explain why the convergence depends upon r and not on c .

Consider the geometric series $\sum_{k = 1}^{\infty} c r^{k}$

Our objective is to prove that convergence depends on r only and not on c .

Step 3. Expansion of the series

$\sum_{k = 1}^{\infty} c r^{k} = c r + c r^{2} + c r^{3} + \dots$ (Series in expanded form )

$= c (r + r^{2} + r^{3} + \dots)$ (Factorize)

$= c \sum_{k = 1}^{\infty} r^{k}$

Thus , if c is any real number then $\sum_{k = 1}^{\infty} {Cr}^{k} = C \sum_{k = 1}^{\infty} r^{*}$ holds .

Therefore the convergence depends only on r .

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Let $\sum_{k = 1}^{\infty} {Cr}^{k}$ be a series with c and $r$ $\in ℝ$ . Explain why the convergence of this series depends only upon the magnitude of r and not c .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explain why the convergence depends upon r and not on c .

Step 3. Expansion of the series

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Let ∑k=1∞ Crkbe a series with c andr∈ℝ. Explain why the convergence of this series depends only upon the magnitude of r and not c .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explain why the convergence depends upon r and not on c .

Step 3. Expansion of the series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Let $\sum_{k = 1}^{\infty} {Cr}^{k}$ be a series with c and $r$ $\in ℝ$ . Explain why the convergence of this series depends only upon the magnitude of r and not c .