show that if lim (n->infinity) (a_n)/(b_n)=0 and ∑ b_n converges then∑ a_n converges?
help me please!
if the two given series of positive terms
a) show that if lim (n->infinity) (a_n)/(b_n)=0 and ∑ b_n (n=1 to infinity) converges then∑ a_n (n=1 to infinity) converges.
b) true or false and why: if lim (n->infinity) (a_n)/(b_n)=0 and ∑ a_n (n=1 to infinity) converges then ∑ b_n (n=1 to infinity) converges.
c) true or false and why: if lim (n->infinity) (a_n)/(b_n)=infinity and ∑ a_n (n=1 to infinity) converges then ∑ b_n (n=1 to infinity) converges.
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1) Since lim(n→∞) a(n)/b(n) = 0, this means that for every ε > 0, we can find a positive integer N such that |a(n)/b(n) - 0| < ε for all n > N.
For concreteness, I'll choose ε = 1:
==> |a(n)| < |b(n)| for all n > N.
Since ∑(n = N+1 to ∞) b(n) converges, we conclude that ∑(n = N+1 to ∞) a(n) also converges by the Direct Comparison Test. Hence, ∑(n = 1 to ∞) a(n) also converges.
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b) False; let a(n) = 1/n^2 (for a convergent p-series), and b(n) = 1/n (for the divergent harmonic series).
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c) True; use part a, and invert the ratio.
I hope this helps!
What does it recommend for a set to converge? follow that to the sequence of partial sums. What does it recommend ? a_n to converge? It follows from that. As for the communicate, evaluate the sequence a_n = a million/n. The reduce of a_n is 0.