Calculus - Sequences and Series

Sequences and Series

Convergence of Sequences

Monotonic Sequence Theorem: Bounded monotonic sequence converges.

Squeeze Theorem: If a_n ≤ b_n ≤ c_n and a_n → L, c_n → L, then b_n → L.

Infinite Series

Definition: $\sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} S_N$ where S_N is partial sum.

Convergence Tests

nth Term Test: If $\lim a_n \neq 0$, series diverges.

Integral Test: If f positive, continuous, decreasing on [1,∞), then ∑a_n converges ⟺ ∫[1 to ∞] f(x)dx converges.

p-Series: ∑1/n^p converges if p > 1.

Comparison Test: If a_n ≤ b_n and ∑b_n converges, then ∑a_n converges.

Limit Comparison Test: If lim(a_n/b_n) = L > 0, then both series converge or both diverge.

Alternating Series Test (Leibniz): If a_n ≥ a_(n+1) and a_n → 0, then ∑(-1)^(n-1)a_n converges.

Ratio Test:

• If lim|a_(n+1)/a_n| < 1: converges absolutely
• If > 1: diverges
• If = 1: inconclusive

Root Test:

• If lim√ⁿ|a_n| < 1: converges absolutely
• If > 1: diverges
• If = 1: inconclusive

Power Series

$f(x) = \sum_{n=0}^{\infty} c_n(x-a)^n$

Radius of Convergence R: Uses ratio or root test.

Interval of Convergence: (a-R, a+R), check endpoints separately.

Taylor Series

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$

Taylor’s Theorem (with Remainder): $f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k + R_n(x)$

where $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$

Common Taylor Series:

• e^x = ∑ x^n/n!
• sin x = ∑ (-1)^n x^(2n+1)/(2n+1)!
• cos x = ∑ (-1)^n x^(2n)/(2n)!
• 1/(1-x) = ∑ x^n for |x| < 1
• ln(1+x) = ∑ (-1)^(n-1) x^n/n