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What is meant by the sequence of partial sums for a seriesk=1ak? Why does it make sense to define the convergence of a series in terms of the convergence of its sequence of partial sums?

Short Answer

Expert verified

The sequence of partial sums of a series k=1ak is the partial sum of the first through nth terms.

As Sn=k=1nak=a1,a1+a2,...,a1+a2+...+an

That's what, it makes a sense to define the convergence of a series in terms of the convergence of its sequence of partial sums.

Step by step solution

01

Step 1. Given information.

Consider the given question,

The series is k=1ak.

02

Step 2. Explain the sequence of partial sums of the given series.

The sequence of partial sums of a series k=1ak is the partial sum of the first through nth terms.

It can be defined as,

Sn=a1+a2+...+an

The sequence of partial sums is given by,

Sn=k=1nak=a1,a1+a2,...,a1+a2+...+an

That's what, it makes a sense to define the convergence of a series in terms of the convergence of its sequence of partial sums.

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