Sequences { Sn } and
Infinite Series ∑ an ;
Finding the Sum of the Series
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Infinite series have an infinite sequence
of partial sums S1 , S2
, S3 , . . . . Sn
associated with the infinite series ∑ an = infinite series =
a1 + a2
+ a3 + . . . + an { an } = infinite sequence of
members of the infinite series
i.e. a1 , a2
, a3 , . . . an { Sn } =
infinite sequence of partial sums for the infinite series S1 = a1, S2
= a1 + a2 |
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Convergence of an Infinite
Series ∑ an and finding its sum An infinite series ∑ an converges
with sum S provided that the limit of its sequence of partial
sums Sn lim Sn
exists and is finite. n →∞ One method to find the sum
of an infinite series is then to find its nth partial sum Sn and proceed to take the
limit of it as n → ∞ . It might not be easy to
find the nth partial sum Sn of an
infinite series. |
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Expressions for the nth
partial sum of a geometric series
∑ rn
can be determined using
the following formulas.
n If r ≠ 1, then the nth partial
sum ∑ ri =
(rn+1 – 1)/(r – 1)
= Sn i
= 0
n Also, if r ≠ 1, then the nth partial
sum ∑ ri =
r(rn – 1)/(r – 1)
= Sn
i = 1 |
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Sum of a Geometric Series If
|r| < 1, then
the geometric series converges. ∞ ∑ a rn
= a + ar + ar2
+ ar3 + . . . = a(1 + r + r2 + r3
+ . . . )
n = 0 ∞ Sum of geometric
series = S
= ∑ a rn =
a/(1 – r) n = 0 If |r|
> 1, then the geometric series diverges. |
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Copyright © 2018 Richard C. Coddington
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