Review Sheet for Calculus 2 Sequences and Series
SEQUENCES
Convergence
A sequence {an}converges if lim anexists and is finite.
Squeeze theorem
If bn≤an≤cnfor all values of n, and lim bn= lim cn=L,
then it implies that lim an=L.
Other Useful facts
anconverges to zero if and only if |an|also converges to zero.
When nis large, ln(n)< np< an< n!< nn
SERIES
Partial sums
sN=
N
X
n=1
an
Convergence
A series is convergent when the limit of partial sums exists,
Xan= lim
N→∞
sN
otherwise it is divergent.
A series is absolutely convergent when P|an|is convergent.
A series is conditionally convergent when P|an|is divergent
but Panis convergent.
Geometric series
Parn−1converges when |r|<1, otherwise diverges.
When convergent, the sum is equal to a
1−r.
p-series
X1
npconverges when p > 1, otherwise diverges.
Divergence Test
If lim an6= 0, then the series Panis divergent.
Integral Test
If an=f(n) when f(x) is a positive, continuous, eventually
decreasing function, then
∞
X
n=1
anconverges ⇐⇒ Z∞
1
f(x)dx converges
Comparison Test
Suppose anand bnare two positive sequences, with an≤bn
for all n > N for some number N.
If Pbnis convergent, then so is Pan.
If Panis divergent, then so is Pbn.
Limit Comparison Test
Suppose that anand bnare two positive sequences, and
lim an
bn=c.
•If c > 0 is a finite number, then
Xanconverges ⇐⇒ Xbnconverges.
•If c= 0 and Pbnconverges, then Panconverges.
•If c=∞and Pbndiverges, then Pandiverges.
Ratio Test
Suppose that lim
an+1
an
=L.
•If L < 1, then Panis absolutely convergent.
•If L > 1, then Panis divergent.
•If L= 1, then the test is inconclusive.
Root Test
Suppose that lim |an|1/n =L.
•If L < 1, then Panis absolutely convergent.
•If L > 1, then Panis divergent.
•If L= 1, then the test is inconclusive.
Alternating Series Test
For series of the form P(−1)nbn, where bnis a positive and
eventually decreasing sequence, then
X(−1)nbnconverges ⇐⇒ limbn= 0
POWER SERIES
Definitions
∞
X
n=0
cnxnOR
∞
X
n=0
cn(x−a)n
Radius of convergence: The radius is defined as the number R
such that the power series converges if |x−a|< R, and diverges
if |x−a|> R.
Interval of convergence:I= interval of values of xfor which
the power series is convergent. Note that the length of the
interval is twice the radius of convergence..
MacLaurin Series
f(x) =
∞
X
n=0
f(n)(0)
n!xn
Taylor Series
f(x) =
∞
X
n=0
f(n)(a)
n!(x−a)n
