Question

Consider the series ∑*n*=1 ∞ *a**n*
where

*a**n*=(5*n*+5)^(9*n*+1)/
12^*n*

In this problem you must attempt to use the Ratio Test to decide
whether the series converges.

Compute

*L*= lim *n*→∞
∣∣∣*a**n*+1/*a**n*∣∣

Enter the numerical value of the limit L if it converges, INF if
the limit for L diverges to infinity, MINF if it diverges to
negative infinity, or DIV if it diverges but not to infinity or
negative infinity.

*L*=

Which of the following statements is true?

**A**. The Ratio Test says that the series converges
absolutely.

**B**. The Ratio Test says that the series
diverges.

**C**. The Ratio Test says that the series converges
conditionally.

**D**. The Ratio Test is inconclusive, but the series
converges absolutely by another test or tests.

**E**. The Ratio Test is inconclusive, but the series
diverges by another test or tests.

**F**. The Ratio Test is inconclusive, but the series
converges conditionally by another test or tests.

Enter the letter for your choice here: ?

Answer #1

Determine whether the series
Summation from n equals 0 to infinity e Superscript negative 5
n∑n=0∞e^−5n
converges or diverges. If it converges, find its sum.
Select the correct choice below and, if necessary, fill in the
answer box within your choice.
A.The series converges because
ModifyingBelow lim With n right arrow infinitylimn→∞
e Superscript negative 5 ne−5nequals=0.
The sum of the series is
nothing.
(Type an exact answer.)
B.The series diverges because it is a geometric series with
StartAbsoluteValue r...

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

Find a general term (as a function of the variable n) for the
sequence{?1,?2,?3,?4,…}={45,1625,64125,256625,…}{a1,a2,a3,a4,…}={45,1625,64125,256625,…}.
Find a general term (as a function of the variable n) for the
sequence {?1,?2,?3,?4,…}={4/5,16/25,64/125,256/625,…}
an=
Determine whether the sequence is divergent or convergent. If
it is convergent, evaluate its limit.
(If it diverges to infinity, state your answer as inf . If it
diverges to negative infinity, state your answer as -inf . If it
diverges without being infinity or negative infinity, state your
answer...

Test the series for convergence or divergence.
∞
(−1)n
8n − 5
9n + 5
n = 1
Step 1
To decide whether
∞
(−1)n
8n − 5
9n + 5
n = 1
converges, we must find lim n → ∞
8n − 5
9n + 5
.
The highest power of n in the fraction is
1
1
.
Step 2
Dividing numerator and denominator by n gives us lim n
→ ∞
8n − 5
9n +...

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?

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...

(1 point) The three series ∑An, ∑Bn, and ∑Cn have terms
An=1/n^8,Bn=1/n^5,Cn=1/n. Use the Limit Comparison Test to compare
the following series to any of the above series. For each of the
series below, you must enter two letters. The first is the letter
(A,B, or C) of the series above that it can be legally compared to
with the Limit Comparison Test. The second is C if the given series
converges, or D if it diverges. So for instance,...

1. Test the series below for convergence using the Root
Test.
∞∑n=1 (4n/10n+1)^n
The limit of the root test simplifies to lim n→∞ |f(n)| where
f(n)=
The limit is:
Based on this, the series
Diverges
Converges
2. We want to use the Alternating Series Test to determine if
the series:
∞∑k=4 (−1)^k+2 k^2/√k^5+3
converges or diverges.
We can conclude that:
The Alternating Series Test does not apply because the terms of
the series do not alternate.
The Alternating Series Test...

1. To test the series ∞∑k=1 1/5√k^3 for convergence, you can use
the P-test. (You could also use the Integral Test, as is the case
with all series of this type.) According to the P-test:
∞∑k=1 1/5√k^3 converges
the P-test does not apply to ∞∑k=1 1/5√k^3
∞∑k=1 1/5√k^3 diverges
Now compute s4, the partial sum consisting of the first 4 terms
of ∞∑k=1 1 /5√k^3:
s4=
2. Test the series below for convergence using the Ratio
Test.
∞∑n=1 n^5 /1.2^n...

1. Test the series below for convergence using the Root
Test.
∞∑n=1 (2n^2 / 9n+3)^n
The limit of the root test simplifies to limn→∞|f(n)|limn→∞|f(n)|
where
f(n)=
2. Test the series below for convergence using the Root
Test.
∞∑n=1 (4n+4 / 5n+3)^n
The limit of the root test simplifies to limn→∞|f(n)| where
f(n)=
The limit is:

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