Answer
Please see step-by-step
Work Step by Step
(a)
The infinite series $\displaystyle \sum_{n=1}^{\infty}a_{n}$
diverges if the sequence of partial sums $\{S_{n}\}$ diverges.
$ S_{n}=a_{1}+a_{2}+\cdot \cdot +a_{n}$ gets arbitrarily large in magnitude (positive or negative).
The sum of a finite number of terms is a finite number A.
Let k be the index of the last deleted term ( the deleted terms are among
$\{a_{1},a_{2},...,a_{k}\}$.
Then for $n\geq k$, the new partial sums are
$(S_{k+1}-A),(S_{k+2}-A),,(S_{k+3}-A)...$
which are obtained by subtracting a finite number A from numbers that grow in magnitude without bound.
This sequence diverges, so the new series will diverge.
(b)
Let S be the sum of the series.
Let k be the index of the last deleted term ( the deleted terms are among
$\{a_{1},a_{2},...,a_{k}\}$.
As in part (a), observe, for $n\geq k$, the new partial sums :
$(S_{k+1}+A),(S_{k+2}+A),(S_{k+3}+A)...$
This seqence is obtained by adding a finite number A to terms that converge to S.
The sequence converges to S+A, so the new series will converge.
