Question

1. Write a geometric series with a starting index of n = 0 which has a sum of 4. Show how you come up with your series.

2. Using the geometric series you wrote in 1, show how you would find the sum if the starting index was n = 3.

3. Write a geometric series that diverges.

Answer #1

1. Let us consider the geometric series

then we should find r such that

Since its a infinte geometric series sum will be

Therefore the series is

3. The series diverges since this is a geometric series with common ratio . Hence it diverges

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...

1. The Taylor series for f(x)=x^3 at 1 is ∞∑n=0 cn(x−1)^n.
Find the first few coefficients.
c0=
c1=
c2=
c3=
c4=
2. Given the series:
∞∑k=0 (−1/6)^k
does this series converge or diverge?
diverges
converges
If the series converges, find the sum of the series:
∞∑k=0 (−1/6)^k=

1. Consider the geometric series −1/4+3/16−9/64+27/256+... Use
summation notation to write this series and determine whether it
converges. If it does, find the sum.

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...

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 the Sum of a Convergent Series:
a.) ∑∞?=0 (-4/9)n
b.) ∑∞?=0 1/(n+1)(n+4)

a) Is the following a geometric series??? Why? b) If it is,
determine if it is convergent or divergent. Why? c) If it is
convergent, find its sum; d) write the next three terms.
1/8 + 1/4 + 1/2 + 1...

rewrite the geometric series 1/8+2/80+4/800+8/8000+.....with
summation notation and determine if it converges or diverges if the
series converge find the value that it converges to.

State whether the given series converges or diverges, and
why.
#21 sum 1/n^5, n=1 to infinity
#22 sum 1/5^n, n=0 to infinity
#23 sum 6^n / 5^n, n=0 to infinity
#24 sum n^-4, n=1 to infinity
#25 sum sqrt(n), n=1 to infinity

Which of the following statements is true?
a) The geometric series Σ∞n=1 r^n is always convergent.
b) for the series Σ∞n=1an, if lim n→∞ an = 1/3, then the series
will be convergent.
c) If an> bn for all values of n and Σ∞n=1 bn is convergent,
then Σ∞ n=1an is also convergent.
d) None of the above

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