Question

Suppose X∞ n=0 c_{n}x^{n} converges when x = −4
and diverges when x = 6. State if the following series are
convergent or divergent, or if there is not information to
determine convergence or divergence.

(a) X∞ n=0 c_{n}3^{n}

(b) X∞ n=0 c_{n}5^{n}

(c) X∞ n=0 c_{n}7^{n}

Answer #1

,

Determine whether the limit converges or diverges, if it
converges, find the limit.
an = (1+(4/n))^n

Determine if the series converges conditionally, converges
absolutely, or diverges.
/sum(n=1 to infinity) ((-1)^n(2n^2))/(n^2+4)
/sum(n=1 to infinity) sin(4n)/4^n

Determine whether the sequence a_n = (3^n + 4^n)^(1/n) diverges
or converges

Determine if each of the following series converges or diverges
showing all the work including all the tests used. Find the sum if
the series converges.
a. Σ (n=1 to infinity) (3^n+1/ 7^n)
b. Σ (n=0 to infinity) e^n/e^n + n

Given that 1/ 1 − x = ∑ n = 0 x^n with convergence in (−1, 1),
find the power series for 1 x with center 5. ∞ ∑ n = 0 Identify its
interval of convergence. The series is convergent from x = , left
end included (enter Y or N): to x = , right end included (enter Y
or N):

Use the RATIO test to determine whether the series is convergent
or divergent.
a) sigma from n=1 to infinity of (1/n!)
b) sigma from n=1 to infinity of (2n)!/(3n)
Use the ROOT test to determine whether the series converges or
diverges.
a) sigma from n=1 to infinity of
(tan-1(n))-n
b) sigma from n=1 to infinity of ((-2n)/(n+1))5n
For each series, use and state any appropriate tests to decide
if it converges or diverges. Be sure to verify all necessary...

Apply term-wise integration to the expansion 1/(1 − x) = ∑∞ n=0
x n = 1 + x + x^2 + x^3 + ... to prove that for −1 < x < 1, −
ln(1 − x) = ∑∞ n=0 (x^n+1)/(n + 1) = x + x^2/2 − x^3/3 + x^4/4 +
... You should find a constant that appears when you integrate. (b)
Study convergence of this new series at the end points of the
interval (−1, 1). (c)...

6. Let series {an} = 1/(n2 + 1) and series {bn} = 1/n2.
Use Limit Comparison Test to determine if each series is convergent
or divergent.
7. Use Ratio Test to determine if series {an}= (n +
2)/(2n + 7) where n is in interval [0, ∞]
is convergent or divergent. Note: if the test is
inconclusive, use n-th Term Test to answer the
question.
8. Use Root Test to determine if series {an} = nn/3(1 +
2n) where n...

Use the ratio test to determine whether∑n=12∞n2+55n
converges or diverges.
(a) Find the ratio of successive terms. Write your
answer as a fully simplified fraction. For n≥12,
limn→∞∣∣∣an+1an∣∣∣=limn→∞
(b) Evaluate the limit in the previous part. Enter ∞
as infinity and −∞ as -infinity. If the limit does
not exist, enter DNE.
limn→∞∣∣∣an+1an∣∣∣ =
(c) By the ratio test, does the series converge,
diverge, or is the test inconclusive?

c.) Determine whether the seriesX∞ k=1 k(k^4 + 2k)/(3k 2 − 7k^5)
is convergent or divergent. If it is convergent, find the sum.
d.) Determine whether the series X∞ n=1 n^2/(n^3 + 1) is
convergent or divergent.

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