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Sec 8.2 Series Part 1
A series is simply the sum of the terms of a sequence.
1 2 3 4 1
n n
a a a a a
∞
=
∑ =^ +^ +^ +^ +"^ is called an^ infinite series^ where a^ n^ is the n^ th^ term of the series.
Associated with any infinite series is a sequence (list), {sn } = s 1 , s 2 , s 3 , s 4 , " , of partial sums where :
s 1 = a 1 , s 2 = a 1 + a 2 , s 3 = a 1 + a 2 + a 3 , s 4 = a 1 + a 2 + a 3 + a 4 ,
s (^) k = a 1 + a 2 + a 3 + a 4 + " + a (^) k
Given a series (^) 1 2 3 4 1
n n
a a a a a
∞
=
∑ =^ +^ +^ +^ +"^ ,^ let^ s^ n^ denote its nth partial sum :^1
1
n n i n i
s a a a a
Partial Sums Test:
If the sequence {sn } is convergent and lim (^) n n s s →∞
= exists as a real number, then the series ∑ an is
convergent and we write 1
n n
a s
∞
=
∑ =.^ The number s is the sum of the series. If the limit does not exist, the
series is divergent.
Telescoping Series: The terms of a series cancel in pairs. Hence the sum collapses into just two terms. Use the partial sums test and look at the limit of those two terms to determine convergence.
EX 1
1
n ln(^ n^ 2)^ ln(^ n 1)
∞
=
⎝ +^ + ⎠
Properties of Series:
1
n n
c a
∞
=
∑ ⋅ =^
1
n n
c a
∞
=
∑. If^
1
n n
a
∞
=
∑ diverges, then^
1
n n
c a
∞
=
∑ diverges.
1
n n n
a b
∞
=
∑ ± =^
1
n n
a
∞
=
∑ ±^
1
n n
b
∞
=
∑. If both^
1
n n
a
∞
=
∑ and^
1
n n
b
∞
=
∑ both converge, then^ (^ )
1
n n n
a b
∞
=
∑ ± converges. If
either 1
n n
a
∞
=
∑ or^
1
n n
b
∞
=
∑ converges and the other diverges, then^ (^ )
1
n n n
a b
∞
=
∑ ± diverges. If both^
1
n n
a
∞
=
∑ and
1
n n
b
∞
=
∑ diverge, you can't tell if^ (^ )
1
n n n
a b
∞
=
∑ ± converges or diverges.
Geometric Series: A series of the form^1 1
n n
ar
∞ − =
∑ = a + ar + ar^
(^2) + ar 3 + ar 4 + " , where a and r are real
numbers with a ≠ 0, is called a geometric series.
Convergence/Divergence of a Geometric Series
The series^1 1
n n
ar
∞ − =
∑ converges to^ 1
a − r
if |r| < 1 and diverges if |r| ≥ 1.
EX 2 Show the following series are geometric series. Then test for convergence. If convergent, find the sum.
A. 2 + 4 +8+ 16 + 32 + "
B.
0
n n n
∞
=
C. 1
n n n
∞ e − =
