Q. 4 - Calculus

Question

What is meant by the tail of a series? Why does it make sense that the tail of a series determines the convergence of the series?

Step-by-Step Solution

Answer

For given $\epsilon >0$, there exists a positive integer N such that $\left|S_k-L\right|<\epsilon$ for $k\ge N$.

By definition of convergence, only the terms after N that depends on the given $\epsilon >0$ determine the convergence. These are the terms that define the tail of the sequence.

Thus, the convergence of a series depends only on the convergence of the tail of the series.

Step 1. Explain the sequence of partial sums of the given series.

Consider the convergent series $\sum _{k=1}^{\infty } a_k$converging to limit L.

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 is given below,

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

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 2. Explain the sequence is converging to limit L.

The sequence $\left\{S_n\right\}$ is converging to limit L.

By definition,

For given $\epsilon >0$, there exists a positive integer N such that $\left|S_k-L\right|<\epsilon$ for $k\ge N$.

Therefore, by definition of convergence, only the terms after N that depends on the given $\epsilon >0$ determine the convergence. These are the terms that define the tail of the sequence.

Thus, the convergence of a series depends only on the convergence of the tail of the series.