Math 521 Midterm 3

series convergence theorem

∑(a_k) converges IFF ∀ε>0 ∃N∈N s.t. ∀n≥m≥n |∑(a_k)|<ε

definition of convergence of an infinite series

if s_n=∑_k=1ⁿ(a_k) [sequence of partial sums] converges to s, then ∑_k=1^∞(a_k) [infinite series] converges to s

the divergence test

if lim(a_k)≠0 then ∑_k=1^∞(a_k) diverges (note: not IFF; consider harmonic series ∑(1/k))

series convergence with boundedness

a series with nonnegative terms converges iff the sequence of partial sums (s_n) is bounded

basic comparison test

Let N₀∈N; 1) Suppose |a_n|≤c_n ∀n≥N₀ then if ∑(c_k) converges, ∑|a_k| and ∑(a_k) converge; 2) Suppose a_n≥d_n≥0 ∀n≥N₀ then if ∑(d_k) diverges, ∑(a_k) diverges

Algebraic Limit Theorem for sequences

1) if ∑a_k converges and c∈R then ∑ca_k=c∑a_k; 2) if ∑a_k converges and ∑b_k converges then ∑(a_k+b_k) = ∑a_k + ∑b_k

Limit Comparison Test

Suppose x_n and y_n are positive sequences and lim(x_n/y_n)=r for r≥0 exists. Then, 1) if r>0 then ∑x_k converges IFF ∑y_k converges; 2) if r=0 then ∑y_k converges IFF ∑x_k converges

absolute convergence def

∑a_k converges absolutely if ∑|a_k| converges

conditionally convergent

∑a_k converges but NOT absolutely

re-arrangement def

∑b_k is a re-arrangement of ∑a_k if there exists a bijection f:N→N s.t. ∀k b_k=a_f(k)

Root Test

Let a_k be a sequence of nonnegative real numbers. Suppose lim(ⁿ√a_n)=r. Then, r<1 implies ∑a_k converges; r>1 implies ∑a_k diverges; r=0 is inconclusive

When to use which test?

ALWAYS start with divergence; for powers use root test; for factorials use ratio test; for all else use BCT

Ratio Test

Let a_k be a sequence of positive real numbers. Suppose [(a_n+1)/(a_n)] converges to λ for some λ∈[0,∞). Then, λ<1 implies ∑a_k converges; λ>1 implies ∑a_k diverges; λ=1 is inconclusive