If∑k=1∞ak and ∑k=1∞bk are convergent series and c is any real number ,then a)∑k=1∞cak=c∑k=1∞akb)∑k=1∞(ak+bk)=∑k=1∞ak+∑k=1∞bk

Q.1 - Calculus | StudyQuestionHub

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) $\sum_{k = 1}^{\infty} (a_{k} + b_{k}) = \sum_{k = 1}^{\infty} a_{k} + \sum_{k = 1}^{\infty} b_{k}$

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Answer

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

1the 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 )