Calc II Sequences & Series

Convergence and Divergence Tests

geometric sequence

a_n = ar^n-1

sum series

∑1/(n+1)(n+a+1)

geometric series test

∑ar^n-1 converges if and only if |r|<1

divergence test (n^th term test)

if lim_n→∞(a_n) ≠ 0…

then ∑_n=1(a_n) diverges

integral test

if f(n) = a_n for all & if f(x) decreases to 0…

 ∑_n=1(a_n) converges if and only if ∫f(x)dx converges

the harmonic series

1/n

p-test

∑_n=1(1/n^p) converges if and only if p > 1

convergence test (direct comparison test)

1. if a_n >= 0 for all n

2. & b_n >= a_n for all n (b_n dominates a_n)

3. & ∑b_n converges, so does ∑a_n…

then ∑a_n converges!

limit comparison test (LCT)

1. if a_n > 0 & b_n > 0 for any n

2. & lim_n→∞(a_n/b_n) converges to a non-zero value…

then ∑_n=1(a_n) & ∑_n=1(b_n) either BOTH converge or BOTH diverge

ratio test

have ∑_n=1(a_n), with a ≠ 0 & any n…

put ρ = lim_n→∞|(a_n+1)/(a_n)|

• then if p > 1, ∑_n=1(a_n) diverges

• if p = 1, we don’t know

• if p < 1, ∑_n=1(a_n) converges

root test (n^th root test)

if a_n > 0, look at lim_n→∞(n√a_n)

• if lim DNE, we don’t know

• if lim < 1, the series converges

• if lim = 1, we don’t know

• if lim > 1, the series diverges

alternating series test

if {|a_n|}_n=1 decreases to 0 & {|a_n|}_n=1 alternates (positive to negative), then ∑_n=1(a_n) converges

• if ∑_n=1(a_n) converges but ∑_n=1|a_n| diverges, we say the series converges CONDITIONALLY!

• if ∑_n=1|a_n| converges, we say the series converges ABSOLUTELY!

error estimate for converging alternating series

power series

a power series is a series in the form ∑_n=1(a_n)(x-x_a)ⁿ

1. apply ratio test to find the radius of convergence,

2. check the endpoints to find the interval of convergence!