relationship between convergence of a sequence and its corresponding series.

Question

I'm preparing for my calculus exam and I'm unsure how to approach these type of questions .

• If the sequence $a_n$ is convergent/divergent what can we about the corresponding series $\sum_{n}a_n$? Is it convergent or divergent?
• If the series $\sum_n a_n$ is convergent/divergent what can we say about the corresponding sequence $a_n$ ? Is it convergent or divergent?

In general, I want to know if there is any kind of relation between the convergence/divergence of a sequence and its corresponding series, and viceversa.

I am also interested if there are some special cases like if series is absolutely convergent or conditionally convergent, then what happens to the corresponding sequence.

It will be a great help answering the question.

P.S- by corresponding series I mean for example the sequence $a_n=n$ has corresponding series $\sum_n n$.

Answer 1

If a sequence $a_n$ which does not converge to $0$, then the series $\sum_{i=1}^n a_n$ does not converge.

If the series $\sum_{i=1}^n a_n$ converges, the sequence $a_n$ must converge to $0$.

These are both kind of obvious. In general, we cannot say that if $a_n$ converges to $0$, then the corresponding series converges (consider $a_n=1/n$). If the series is absolutely or conditionally convergent, the result still holds, clearly.

We can use the comparison test on $a_n$ to see if its series converges. For instance, if $a_n\leq 1/n^2$, then $\sum_{i=1}^n a_n$ must converge.

Answer 2

Let $(a_j)_{j=0}^\infty$ be a sequence, and $\sum_{j=0}^\infty a_j$ the corresponding series.

• Convergent sequence, convergent series: Set $a_j = 0$.

• Convergent sequence, divergent series: Set $a_j = \frac{1}{j + 1}$.

• Divergent sequence, convergent series: Does not exist. If $\sum_{j=0}^\infty a_j$ converges, then $a_j \to 0$.

• Divergent sequence, divergent series: Set $a_j = (-1)^j$.

In general, the only relationship that can be drawn is $\sum_{j=0}^\infty a_j$ convergent $\implies$ $a_j$ converges to $0$. It also holds if the series is absolutely convergent/conditionally convergent.