Question

How do I find the value of the convergent series?

Answer 1

\[\sum_{n=1}^{\infty} \frac{ 1 }{ n(n+6)}\]

Answer 2

i would try partial fractions

Answer 3

and see if the series telescopes

Answer 4

im not sure how to do that.. :(

Answer 5

hmm... that is the only way I know... you sure you guys haven't talked about telescoping series?

Answer 6

i don't think i've learned about that

Answer 7

what have you guys mentioned in class

Answer 8

the topic is divergence and integral test

Answer 9

but those can't use to find the value of the series

Answer 10

oh...

Answer 11

do you know at least how to write your fraction as a sum of partial fractions ?

Answer 12

ok, i have 20 more min to do my online hw so can I show you another problem that might be easier?

Answer 13

i guess... so i guess that means you haven't done partial fractions here? Like you might remember them using them to ingrate certain things...

Answer 14

for example
\[\frac{1}{n(n+6)}=\frac{A}{n}+\frac{B}{n+6}\]
and you would combine the fractions on the right to find A and B

Answer 15

Test each of the following series for convergence by the Integral Test. If the integral test cannot be applied to the series, enter NA.\[\sum_{n=1}^{\infty}\frac{ n+5 }{ (-4)^n}\] ∑n=1∞n+5(−4)n

Answer 16

The integral test says:
Suppose that f(n) is continuous, positive, and decreasing function on the interval [1,inf)
then
\[\int\limits_k^ \infty f(x) dx \text{ converges } \implies \sum_{n=k}^{\infty} f(n) \text{ converges } \\ \int\limits_k^\infty f(x) dx \text{ diverges } \implies \sum_{n=k}^{\infty} f(n) \text{ diverges }\]


do you think that we satisfy the hypothesis part of this test ?

Answer 17

\[f(x)=\frac{x+5}{(-4)^{x}} \\ \text{ is this continuous on } [1,\infty) ? \\ \text{ is this positive on } [1,\infty)? \\ \text{ does this decrease on } [1,\infty)?\]

Answer 18

its decreasing, continuous, but not positive?

Answer 19

increasing

Answer 20

it is also not continuous
for example at x=1/2 the denominator is not defined over real numbers

but anyways we just need it to fail at least one of those 3 things to rule out the integral test


and you definitely got that it wasn't completely positive on [1,inf)

Answer 21

ahh ok.

Answer 22

that's it?

Answer 23

it is your question :p
It says use the integral test if it can be applied. If it cannot be applied then say NA.

Answer 24

was there suppose to be more to the question? because that is all I see...

Answer 25

ratio test might work here
but we cannot use the integral test because it didn't meet the conditions to apply the integral test

Answer 26

or i mean alternating series test

Answer 27

ok thank you !