29. Series - first results and a first test for convergence

Relation between series and sequences

Consider the series

$$\sum _{k=1}^{\infty }{a}_k$$

If ${\sum }_{k=1}^{\infty }$ converges by assumption then

$$a_k\to 0\text{ as }k\to \infty$$

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Showing that a series diverges

If you can show that

$$a_k\nrightarrow 0\text{ as }k\to \infty$$

Then the series $s_n$ doesn’t converge

Proof

Let $s_n={\sum }_{k=1}^n{a}_k$, then $(s_n)$ converges by assumption

Suppose $s_n\to s$ as $n\to \infty$ then

$$s_{n-1}\to s\text{ as }n\to \infty$$

So by AOL

$$a_n=s_n-s_{n-1}\to s-s=0\text{ as }n\to \infty$$

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Positive sequence with bounded series converges

Let $(a_k)$ be a sequence of non-negative real numbers
Let $s_k={\sum }_{k=1}^n{a}_k$

Suppose that $(s_n)$ is bounded then

$$\sum _{k=1}^{\infty }{a}_k\text{ converges}$$

Proof

Since $a_k\ge 0$ for all $k$ then $(s_n)$ is increasing
So $(s_n)$ is monotone and bounded, so by Monotone Sequences Theorem

$$(s_n)\text{ converges hence }\sum _{k=1}^{\infty }{a}_k\text{ converges}$$

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Comparison Test

Let $(a_k)$ and $(b_k)$ be real sequences
Assume that

$$0\le a_k\le b_k\text{ for all }k\ge 1$$

and

$$\sum _{k=1}^{\infty }{b}_k\text{ converges}$$

Then

$$\sum _{k=1}^{\infty }{a}_k\text{ converges}$$

Generally it works for $0\le a_k\le C{b}_k\text{ for all }k\ge 1$ for some $C\in \mathbb{R}$

Comparison Test for Divergence

If

$$0\le a_k\le b_k\text{ for all }k$$

and

$$\sum _{k=1}^{\infty }{a}_k\text{ diverges}$$

Then

$$\sum _{i=1}^{\infty }{b}_k\text{ diverges}$$

Proof

Let $s_n={\sum }_{k=1}^n{a}_k$
Then $(s_n)$ is increasing, since $a_k\ge 0$ for all $k\ge 1$
And

$$s_n=\sum _{k=1}^n{a}_k\le \sum _{k=1}^n{b}_k\le \sum _{k=1}^{\infty }{b}_k$$

Since the last series converges then $(s_n)$ is bounded
Hence by Monotone Sequences Theorem

$$\sum _{k=1}^{\infty }{a}_k\text{ converges}$$

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