Question

If
a series is convergent , what must be true of the individual terms
of the series as n tends towards infinity ?

1b)
Is every absolutely convergent series also convergent ? Justify
your ans

Answer #1

Let Sum from n=1 to infinity (an) be a convergent
series with monotonically decreasing positive terms. Prove that
limn→∞ n(an) = 0.

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

Give an example of a convergent alternating series where the
conditions of the alternating series test do not hold. You don’t
need to give an explicit formula for the terms of the series. Just
describe it in words if you prefer. Carefully justify your
answer.

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...

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...

True or False. If all terms of a series are positive, the series
sums to a positive number. Justify your answer.

Question D:
i) Write out the first few terms of the series sum 9*0.1^n, n=1
to infinity
ii) What is the series equal to?
iii) What famous math “paradox” does this relate to?

For the next two series, (1) find the interval of convergence
and (2) study convergence at the end points of the interval if any.
Also, (3) indicate for what values of x the series converges
absolutely, conditionally, or not at all. You must indicate the
test you use and show the interval of convergence both analytically
and graphically and summarize your results on the picture.
∑∞ n=1 ((−1)^n−1)/ (n^1/4)) *x^n

) Explain in your own words what is wrong with, and how you
might fix, the following statement. “The series from n=1 to
infinity 1/(2n^2−1) is convergent by the Direct Comparison
Test”

How many terms of the series (-1)^n/n! do you need to add up for
the partial sum to be at most 0.00001 away from the true sum of the
series? What is the value of the partial sum?

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