Determine if the following series converge or diverge. If it
converges, find the sum.
a. ∑n=(3^n+1)/(2n) (upper limit of sigma∞, lower limit is
n=0)
b.∑n=(cosnπ)/(2) (upper limit of sigma∞ , lower limit is n=
1
c.∑n=(40n)/(2n−1)^2(2n+1)^2 (upper limit of sigma ∞ lower limit
is n= 1
d.)∑n = 2/(10)^n (upper limit of sigma ∞ , lower limit of sigma
n= 10)
1. Given the series:
∞∑k=1 2/k(k+2)
does this series converge or diverge?
converges
diverges
If the series converges, find the sum of the series:
∞∑k=1 2/k(k+2)=
2. Given the series:
1+1/4+1/16+1/64+⋯
does this series converge or diverge?
diverges
converges
If the series converges, find the sum of the series:
1+1/4+1/16+1/64+⋯=
Use induction to prove
Let f(x) be a polynomial of degree n in Pn(R). Prove that for
any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that
g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the
nth derivative of f(x).
Prove that every sequence in a discrete metric space converges
and is a Cauchy sequence.
This is all that was given to me... so I am unsure how I am
supposed to prove it....