Chapter 7: Q.1 (page 625)

If $\sum_{k = 1}^{\infty} a_{k}$ and $\sum_{k = 1}^{\infty} b_{k}$ are convergent series and c is any real number ,then
a) $\sum_{k = 1}^{\infty} c a_{k} = c \sum_{k = 1}^{\infty} a_{k}$
b) $\sum_{k = 1}^{\infty} (a_{k} + b_{k}) = \sum_{k = 1}^{\infty} a_{k} + \sum_{k = 1}^{\infty} b_{k}$

Short Answer

Expert verified

This means that adding or removing any finite number of terms from the start of the series does not change its convergent series

Step by step solution

the given data

A) since we already know that both limits and sums commute with sums and constant multiples we will prove part b and leave the proof of part a )

B) given information ∑k=1∞(ak+bk)=∑k=1∞ak+∑k=1∞bk

The result follows from looking at the limits of partial sums

$\sum_{k = 1}^{\infty} (a_{k} + b_{K}) = \lim_{n \to \infty} \sum_{k = 1}^{\infty} (a_{k} + b_{k}) = \lim_{n \to \infty} (\sum_{k = 1}^{\infty} a_{k} + \sum_{k = 1}^{\infty} b_{k}) = \lim_{n \to \infty} \sum_{k = 1}^{\infty} a_{k} + \lim_{n \to \infty} \sum_{k = 1}^{\infty} b_{k} = \sum_{k = 1}^{\infty} a_{k} + \sum_{k = 1}^{\infty} b_{k}$

hence proved

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If $\sum_{k = 1}^{\infty} a_{k}$ and $\sum_{k = 1}^{\infty} b_{k}$ are convergent series and c is any real number ,then
a) $\sum_{k = 1}^{\infty} c a_{k} = c \sum_{k = 1}^{\infty} a_{k}$
b) $\sum_{k = 1}^{\infty} (a_{k} + b_{k}) = \sum_{k = 1}^{\infty} a_{k} + \sum_{k = 1}^{\infty} b_{k}$

Short Answer

Step by step solution

the given data

B) given information ∑k=1∞(ak+bk)=∑k=1∞ak+∑k=1∞bk

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If∑k=1∞akand ∑k=1∞bkare convergent series and c is any real number ,thena)∑k=1∞cak=c∑k=1∞akb)∑k=1∞(ak+bk)=∑k=1∞ak+∑k=1∞bk

Short Answer

Step by step solution

the given data

B) given information ∑k=1∞(ak+bk)=∑k=1∞ak+∑k=1∞bk

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Probability and Statistics

Study anywhere. Anytime. Across all devices.

If $\sum_{k = 1}^{\infty} a_{k}$ and $\sum_{k = 1}^{\infty} b_{k}$ are convergent series and c is any real number ,then
a) $\sum_{k = 1}^{\infty} c a_{k} = c \sum_{k = 1}^{\infty} a_{k}$
b) $\sum_{k = 1}^{\infty} (a_{k} + b_{k}) = \sum_{k = 1}^{\infty} a_{k} + \sum_{k = 1}^{\infty} b_{k}$