Calculus Series

Geometric Series

first term/(1-r)

convergent if |r| < 1

divergent if |r| ≥ 1

Basic Divergence Test

If lim(n→∞) a(n) = DNE or ≠ 0 then the series is divergent

Integral Test

ƒ is continuous, positive, decreasing function

on [1, ∞)

if ∫1,∞ f(x)dx is convergent, then ∑ a(n) is convergent

and vice versa

P-Series

∑ 1÷n^p is convergent if p > 1 and divergent if p ≤ 1

Comparison Test

∑a(n) ∑b(n) are series with positive terms

if ∑b(n) is convergent and a(n)≤b(n) for all n, then ∑a(n) is also convergent

if ∑b(n) is divergent and a(n)≥b(n) for all n, then ∑a(n) is also divergent

Limit Comparison Test

∑a(n) ∑b(n) are series with positive terms

if lim(n→∞) (An/Bn) = c

where c is a finite number and c > 0, then both series act the same

Alternating Series Test

if the alternating series ∑(-1)^n*B(n) where B(n) > 0 and the series decreases to 0, then the series is convergent

Absolute Convergence

Series ∑a(n) is absolutely convergent if the series of absolute values ∑ |a(n)| is convergent

Conditional Convergence

Series ∑a(n) is conditionally convergent if it is convergent but not absolutely

The Ratio Test

If lim(n→∞) |a(n+1)/a(n)| = L < 1 then the series ∑a(n) is absolutely convergent

If lim(n→∞) |a(n+1)/a(n)| = L > 1 then the series ∑a(n) is divergent

If lim(n→∞) |a(n+1)/a(n)| = 1 then it's inconclusive

The Root Test

if lim(n→∞) nthroot(|a(n)|) = L < 1 then the series ∑a(n) is absolutely convergent

if lim(n→∞) nthroot(|a(n)|) = L > 1 then the series ∑a(n) is divergent

if lim(n→∞) nthroot(|a(n)|) = 1 then inconclusive

1/(1-x) = ...

= 1 + x + x^2 + x^3+ ... = ∑ x^n

R = 1

e^x = ...

= 1 + x/1! + x^2/2! + x^3/3! + x^4/4! + ... = ∑x^n/n!

R = ∞

sinx = ...

= x - x^3/3! + x^5/5! - x^7/7! + ... = ∑(-1)^n*(x^2n+1)/(2n+1)!

R = ∞

cosx = ...

= 1 - x^2/2! + x^4/4! - x^6/6! + ... = ∑(-1)^n*(x^2n)/(2n)!

R = ∞

arctanx = ...

= x - x^3/3 + x^5/5 - x^7/7 + ... = ∑(-1)^n*(x^2n+1)/(2n+1)

R = 1

ln(1 + x) = ...

= x - x^2/2 + x^3/3 - x^4/4 + ... = ∑(-1)^(n-1)*(x^n)/n

(note, the sigma starts at n=1)

R = 1