Q. 3 - Calculus

Question

What is meant by the sequence of partial sums for a series $\sum _{k=1}^{\infty } a_k$? Why does it make sense to define the convergence of a series in terms of the convergence of its sequence of partial sums? 

Step-by-Step Solution

Answer

The sequence of partial sums of a series $\sum _{k=1}^{\infty } a_k$ is the partial sum of the first through nth terms.

As $$\begin{gathered} \left\{S_n\right\}=\left\{\sum _{k=1}^n a_k\right\} \\ =\left\{a_1,\left(a_1+a_2\right),...,\left(a_1+a_2+...+a_n\right)\right\} \end{gathered}$$

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 1. Given information.

Consider the given question,

The series is $\sum _{k=1}^{\infty } a_k$.

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

The sequence of partial sums of a series $\sum _{k=1}^{\infty } a_k$ is the partial sum of the first through nth terms.

It can be defined as,

$S_n=a_1+a_2+...+a_n$

The sequence of partial sums is given by,

$$\begin{gathered} \left\{S_n\right\}=\left\{\sum _{k=1}^n a_k\right\} \\ =\left\{a_1,\left(a_1+a_2\right),...,\left(a_1+a_2+...+a_n\right)\right\} \end{gathered}$$

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.