Question

Anyone here good with series and the interval of convergence?

Answer 1

I might be okay. It depends.

Answer 2

Power series?

Answer 3

yes

Answer 4

up to diff eq.

Answer 5

This one is giving me a problem. Ok interval of convergence and radius.
6^n(x+5)/(sgrt of n)

Answer 6

The sum of this, infinity bounds, n=1

Answer 7

I did ratio test and got 1, so it is inconclusive... I don't know where to go now haha

Answer 8

Nevermind... I took out the (x+5) when I was doing the limit and forgot about it...

Answer 9

Well, it would be up to you then.

Answer 10

Did you get R=5, (-5, 5)

Answer 11

just for clarity...is this your problem

\[\sum_{n=1}^{\infty}\frac{6^{n}(x+5)}{\sqrt{n}}\]

Answer 12

Yessum

Answer 13

then \(R\neq 5\)

Answer 14

No, the (x+5) is to the n power sorry

Answer 15

ah...that is what i figured

Answer 16

\[\sum_{n=1}^{\infty}\frac{6^{n}(x+5)^{n}}{\sqrt{n}}\]

Answer 17

Ok so the (-5, 5) and R=5 is wrong anyway

Answer 18

Si

Answer 19

Should I use another test?

Answer 20

use the ratio test...it works

Answer 21

I'm getting 1... so it's inconclusive.

Answer 22

then you are not doing it correctly

Answer 23

So when you solved it, did you get that the series converges or doesn't converge?

Answer 24

Merp... I don't see what I am doing wrong :(

Answer 25

\[a_n=\frac{6^{n}(x+5)^{n}}{\sqrt{n}}\]

\[\left|\frac{a_{n+1}}{a_n}\right|=\left|\frac{\frac{6^{n+1}(x+5)^{n+1}}{\sqrt{n+1}}}{\frac{6^{n}(x+5)^{n}}{\sqrt{n}}}\right|\]

\[=\left|\frac{6^{n+1}(x+5)^{n+1}}{\sqrt{n+1}}\frac{\sqrt{n}}{6^{n}(x+5)^{n}}\right|\]


=...

Answer 26

can you finish from here?

Answer 27

\[(6(x+5)\sqrt{n})/\sqrt{n+1}\]

Answer 28

you lost your absolute value

Answer 29

Then I do the limit as n approaches infinity.
I get 6(x+5) lim (sqrt n)/(sqrt n+1)

Answer 30

I mean, what does the absolute value do to the equation? I should still get a positive value. I'm sorry... I pulled an all-nighter to study for my orgo test :( Brain is not working correctly

Answer 31

you need the absolute value...you need to get two answers (so that you can eventually find the interval of convergence)

Answer 32

-5 and 5

Answer 33

no

Answer 34

\[\lim_{n\to\infty}\left|\frac{6(x+5)\sqrt{n})}{\sqrt{n+1}}\right|=|6(x-5)|\]

we then set

\[|6(x+5)|<1\] and solve for \(x\)

Answer 35

Ok...
-29/6 and -31/6
That doesn't look right at all...

Answer 36

those are correct

Answer 37

Ok, so how do you find the radius from that?

Answer 38

\[|6(x+5)|<1\]

so

\[|x+5|<\frac{1}{6}\]

so \[R=\frac{1}{6}\]

Answer 39

Even with the x+5 on the other side?

Answer 40

This is the question I missed on the exam last week... ugh, thank you so much :)

Answer 41

what is 1/2 the distance from -31/6 to -29/6

Answer 42

Ok got it :) Thank you, thank you, thank you!

Answer 43

do you need the interval of convergence?

Answer 44

Shouldn't it be (-31/6) to (-29/6)?

Answer 45

No, it would be a closed bracket for the latter

Answer 46

well ..which one is it

\[\left[\frac{-31}{6},\frac{-29}{6}\right]\]
\[\left(\frac{-31}{6},\frac{-29}{6}\right]\]
\[\left[\frac{-31}{6},\frac{-29}{6}\right)\]
\[\left(\frac{-31}{6},\frac{-29}{6}\right)\]

Answer 47

I get a complete answer for both, so the first one in your list?

Answer 48

no

Answer 49

Sorry, the third down

Answer 50

correct

Answer 51

Can you explain that? I got -1 and 1 with each... so shouldn't they both be in brackets?

Answer 52

plug back into the original sum

Answer 53

for -29/6 you get

\[\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\] which diverges

with -31/6 you get

\[\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}\] which converges by the alternating series test

Answer 54

Oh, ok. I just tried to solve the whole thing as an equation and not another series function. Thank you!

Answer 55

no problem