In: Advanced Math

Consider the power seriesf(z) =∑∞k=0akzkwhereak=(2k,keven,17kkodd.

(a)Show that neither limn→∞an+1annor limn→∞|an|1nexist.

(b)Find the radius of convergence of each power series:f1(z) =∞∑m=022mz2mandf2(z) =∞∑m=0z2m+172m+1.

(c)Verify thatf(z) =f1(z) +f2(z). Use the radii of convergence off1(z)andf2(z)to infer theradius of convergence off(z)

Find Taylor series expansion of log(1+z) and show radius of
convergence

Find the sum and interval of convergence a power series
I need radius of convergence or sum of series

Find the power series representation of the following functions
and find the corresponding radius of convergence.
1. x2(ln(5+ x))
2. 4x/(1 + x)3

Determine the radius of convergence and the interval of
convergence of the following power series.
∞ ∑ n=1
(2^(1+2n))/ (((−3)^(1+2n)) n^2) (4x+2)^n .

Find the radius of convergence, R, of the series. Find
the interval, I, of convergence of the series. (Enter your
answer using interval notation
∞
(−1)n
(x −
4)n
3n +
1
n = 0
∞
(x −
4)n
n7 + 1
n = 0
∞
7n (x +
5)n
n
n = 1
∞
(x −
13)n
nn
n = 1
∞
4nxn
n2
n = 1

find the radius of convergence and interval of convergence of
the series ∑ n=1 ~ ∞ (3^n)((x+4)^n) / √n
Please solve this problem with detailed process of solving.
I can't understand why the answer is [-13/3, -11/3)
I thought that the answer was (-13/3, -11/3].
Can you explain why that is the answer?

Show that a graph with at least 2k vertices is k-connected if and
only if for any two unrelated subsets X, Y of V, such that | X | =
k = | Y |, there are k foreign paths between X and Y.

#13) Find the radius and interval of convergence of the power
series (Sigma∞ n=1) (−1)^n(x − 1)^n/n4^n by responding to the
following sequence of questions.
(a) Compute the limit L = lim n→∞ |an+1|/|an| .
(b) Given that the power series absolutely converges for L <
1 by the Ratio Test, compute the radius of convergence, where the
radius of convergence is the real number R such that the power
series converges for all |x| < R.
(c) Test whether...

Consider the series X∞ k=2 2k/ (k − 1)! . (a) Determine whether
or not the series converges or diverges. Show all your work! (b)
Essay part. Which tests can be applied to determine the convergence
or divergence of the above series. For each test explain in your
own words why and how it can be applied, or why it cannot be
applied. (i) Divergence Test (ii) Direct Comparison test to X∞ k=2
2k /(k − 1). (iii) Ratio Test

1) Find the radius of convergence and interval
of convergence of the given series Σ x^2n/n!
2) Find the power series representation of
f(x)=(x-1)/(x+2) first then find its interval of convergence.

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