How to find the sum of a sequence of partial sums

What is a series of partial sums?

Remember, a normal series is given by

$\sum^{\infty}_{n=1}a_n$

where $a_n$ is a sequence whose $n$ values increase by increments of $1$. For example, this series could be

$\sum^{\infty}_{n=1}a_n=1,\ 2,\ 3,\ 4,\ 5,\ 6,\ ...\ a_n$

Hi! I'm krista.

I create online courses to help you rock your math class. Read more.

On the other hand, a partial sums sequence is called $s_n$, and its $n$ values increase by additive increments. This means that the first term in a partial sums sequence is the $n=1$ term, the second term is the $n=1$ term plus the $n=2$ term, the third term is $(n=1)+(n=2)+(n=3)$, etc.

A normal series is related to its corresponding partial sums sequence by

$\sum^{\infty}_{n=1}a_n=\lim_{n\to\infty}s_n$

This equation is critical, because it allows us to work backwards from the partial sums sequence to the original series, $a_n$.

How to find the sum of a series of partial sums

---

Take the course

Want to learn more about Calculus 2? I have a step-by-step course for that. :)

---

Using a limit to find the sum of the series of partial sums

Example

Find the sum of the sequence of the partial sums.

$s_n=1-2(0.4)^n$

This question is asking us to find the sum of the series $a_n$, given its corresponding sequence of partial sums, so we can use

$\sum^{\infty}_{n=1}a_n=\lim_{n\to\infty}s_n$

$\sum^{\infty}_{n=1}a_n=\lim_{n\to\infty}1-2(0.4)^n$

On the other hand, a partial sums sequence is called s_n, and its n values increase by additive increments.

Now we can evaluate the limit.

$\sum^{\infty}_{n=1}a_n=1-2(0.4)^{\infty}$

When $0.4$ is raised to the power of $\infty$, it’ll become smaller and smaller and eventually approach $0$.

$\sum^{\infty}_{n=1}a_n=1-2(0)$

$\sum^{\infty}_{n=1}a_n=1$

The sum of the series $a_n$ given the sequence of the partial sums $s_n=1-2(0.4)^n$ is $1$.

---

Get access to the complete Calculus 2 course

---