Question

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 does not apply because the absolute value of the terms do not approach 0, and the series diverges for the same reason.
- The series converges by the Alternating Series Test.
- The series diverges by the Alternating Series Test.
- The Alternating Series Test does not apply because the absolute value of the terms are not decreasing.

Answer #1

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:

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. To test this series for convergence
∞∑n=1 n /√n^3+1
You could use the Limit Comparison Test, comparing it to the series
∞∑n=1 1 /n^p where p=
2. Test the series below for convergence using the Ratio
Test.
∞∑n=1 n^5/0.5^n
The limit of the ratio test simplifies to lim n→∞|f(n)| where
f(n)=
The limit is:

Apply the Root Test to determine convergence or divergence, or
state that the Root Test is inconclusive.
from n=1 to infinity (3n-1/4n+3)^(2n)
Calculate lim n→∞ n cube root of the absolute value of an
What can you say about the series using the Root Test?
Determine whether the series is absolutely convergent,
conditionally convergent, or divergent.

Test for absolute or conditional convergence (infinity, n=1)
((-1)^n(4n))/(n^3+1)

Consider the series ∑n=1 ∞ an
where
an=(5n+5)^(9n+1)/
12^n
In this problem you must attempt to use the Ratio Test to decide
whether the series converges.
Compute
L= lim n→∞
∣∣∣an+1/an∣∣
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...

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 +...

Test the series for convergence using the Alternating Series
Test: X∞ m=2 (−1)^m/ (m 2^m). If convergent, determine whether this
series converges absolutely or conditionally

Determine if the series, k = 1 to ∞ , with terms
(-1)k ((k - 1)/(k3/2 + 1))2
converges absolutely, converges conditionally, or diverges. Tell
which convergence test you are using.

The power series ∑ an is defined as
a1 = 1/2 and an+1=((4n-1)/(3n+2))
an . Determine if the series converges or diverges.
Write the test step by step.

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