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???
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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
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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???.
