An Example of the Connection Between Sequences and Series:
From Section 11.2, #55
Question:
Given the partial sum of our series is:
find the sum of the series,
and the terms in our series,
a
n
.
Solution:
We have been given an the partial sums:
So the first part, finding
S
,
is the easy part: just take the limit.
Finding
a
n
isn't that bad either.
Consider our partial sum:
If we look at the partial sum
S
n
−
1
,
we see that it stops with one term earlier, ie. without
a
n
.
So if we subtract:
But we've been given a formula for
S
n
(and thus
S
n
−
1
):
1
1
n
n
S
n
1
i
i
S
a
1
1
1
n
n
i
i
n
S
a
n
1
1
1
1
1
0
lim
lim
lim
1
1
1
1
0
1
i
n
n
n
n
i
n
n
a
S
S
n
n
1
2
3
1
1
.....
n
n
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 Winter '10
 BEN
 Sequences And Series, Summation, Sn, Partial sum

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