Review Sheet for Calculus 2 Sequences and Series

Limit Comparison Test SEQUENCES Suppose that a and b are two positive sequences, and Convergence n n lim an = c. bn A sequence {an} converges if lim an exists and is finite. • If c > 0 is a finite number, then Squeeze theorem X X an converges ⇐⇒ bn converges.

If bn ≤ an ≤ cn for all values of n, and lim bn = lim cn = L, P P then it implies that lim an = L. • If c = 0 and bn converges, then an converges. P P • If c = ∞ and bn diverges, then an diverges. Other Useful facts Ratio Test an converges to zero if and only if |an| also converges to zero.

p n n an+1 When n is large, ln(n) < n < a < n! < n Suppose that lim = L. an P SERIES • If L < 1, then an is absolutely convergent. P Partial sums • If L > 1, then an is divergent. • If L = 1, then the test is inconclusive. N X sN = an n=1 Root Test

1/n Convergence Suppose that lim |an| = L. P A series is convergent when the limit of partial sums exists, • If L < 1, then an is absolutely convergent. P X • If L > 1, then an is divergent. an = lim sN N→∞ • If L = 1, then the test is inconclusive. otherwise it is divergent. P Alternating Series Test A series is absolutely convergent when |an| is convergent. P A series is conditionally convergent when |an| is divergent For series of the form P(−1)nb , where b is a positive and P n n but an is convergent. eventually decreasing sequence, then

Geometric series X n (−1) bn converges ⇐⇒ lim bn = 0 P arn−1 converges when |r| < 1, otherwise diverges. a When convergent, the sum is equal to . 1 − r POWER SERIES p-series Definitions X 1 ∞ ∞ p converges when p > 1, otherwise diverges. n X n X n cnx OR cn(x − a) Divergence Test n=0 n=0 Radius of convergence: The radius is defined as the number R If lim a 6= 0, then the series P a is divergent. n n such that the power series converges if |x−a| < R, and diverges if |x − a| > R. Integral Test Interval of convergence: I = interval of values of x for which If a = f(n) when f(x) is a positive, continuous, eventually n the power series is convergent. Note that the length of the decreasing function, then interval is twice the radius of convergence.. ∞ X Z ∞ an converges ⇐⇒ f(x)dx converges MacLaurin Series n=1 1

∞ (n) Comparison Test X f (0) n f(x) = x n! n=0 Suppose an and bn are two positive sequences, with an ≤ bn for all n > N for some number N. Taylor Series P P If bn is convergent, then so is an. ∞ (n) X f (a) n f(x) = (x − a) P P n! If an is divergent, then so is bn. n=0