Question

Consider the power series (n^10(x+7)^n)/((5^n)(n^(32/3)))
need to find the interval of convergence and radius of convergence, please help

Answer 1

Have you tried applying the ratio test?

Answer 2

i have but am not fully understand it yet

Answer 3

this is what we want:
\[|\frac{a_{n+1}}{a_n}|<1 \]

well first we can simplify the left hand expression a bit
\[|a_{n+1} \cdot \frac{1}{a_n}|=|\frac{(n+1)^{10} (x+7)^{n+1}}{5^{n+1}( n+1)^\frac{32}{3}} \cdot \frac{5^n n^\frac{32}{3}}{n^{10}}(x+7)^n | \]

Answer 4

for example
\[\frac{5^n}{5^{n+1}}=\frac{1}{5} \\ \frac{(x+7)^{n+1}}{(x+7)^n}=x+7\]
and yes that (x+7)^n was suppose to be in the bottom

Answer 5

\[|a_{n+1} \cdot \frac{1}{a_n}|=|\frac{(n+1)^{10}(x+7)^{n+1}}{5^{n+1}(n+1)^\frac{32}{3}} \cdot \frac{5^n n^\frac{32}{3}}{n^{10} (x+7)^n}| \\ |\frac{a_{n+1}}{a_n}|=\frac{1}{5}|x+7| \cdot |\frac{(n+1)^{10} n^\frac{32}{3}}{(n+1)^\frac{32}{3}n^{10}}|\]
and we have n goes to infinity

Answer 6

so you need to find the limit of that thing as n goes to infinity
and we want that to be less than 1

Answer 7

okay thanks, can u show me the answer or show me who to get there?

Answer 8

to find the interval of convergence and radius of convergence for it

Answer 9

\[\frac{1}{5} |x+7| | \lim_{n \rightarrow \infty } \frac{(n+1)^{10}n^\frac{32}{3}}{(n+1)^{\frac{32}{3}}n^{10}}|<1\]
well you first need to evaluate that limit I was talking about
then you can solve for x

Answer 10

so just look at1/5(x+7) how do u eval limit?

Answer 11

are you saying you don't know how to evaluate this limit?
\[\lim_{n \rightarrow \infty} \frac{(n+1)^{10} n^{\frac{32}{3}}}{(n+1)^\frac{32}{3}n^{10}}\]

Answer 12

\[\lim_{n \rightarrow \infty} \frac{n^{\frac{32}{3}-10}}{(n+1)^{\frac{32}{3}-10}} \\ =\lim_{n \rightarrow \infty} (\frac{n}{n+1})^\frac{2}{3} \\ = (\lim_{n \rightarrow \infty}\frac{n}{n+1})^\frac{2}{3}\]
can you evaluate the limit now ?

Answer 13

is it 1?

Answer 14

yes the inside thing goes to 1
and 1^(2/3) is 1

Answer 15

\[\frac{1}{5} |x+7| | \lim_{n \rightarrow \infty } \frac{(n+1)^{10}n^\frac{32}{3}}{(n+1)^{\frac{32}{3}}n^{10}}|<1 \\ \frac{1}{5} |x+7| |1|<1 \\ \frac{1}{5} |x+7|<1\]

Answer 16

is this for R or X?

Answer 17

is x then 35?

Answer 18

what does that mean ?
Solve for x to find the interval of convergence
you might want to consider testing the endpoints since at L=1 the ratio test is inconclusive
I don't see how you got x=35 when we had an inequality to solve.

Answer 19

recall |f|<a
where a is positive you have
-a<f<a

Answer 20

1/5|x+7|<1
|x+7|<5
-5<x+7<5
can you solve this inequality?

Answer 21

-12 for both?

Answer 22

-5-7 is -12
but I don't understand how 5-7 is -12 also

Answer 23

5-7 should be -2
since we have 5-7=-(7-5)=-(2)=-2

Answer 24

oh okay so -2 and -12 for x, and how to find R?

Answer 25

\[-5<x+7<5 \\ \text{ subtract 7 } \\ -5-7 <x<5-7 \\ -12<x<-2 \\ \\ \text{ testing the endpoints we will see that the series still diverges so} \\ \text{ the interval of convergence is } -12<x<-2 \\ \text{ anyways the diameter is } -2-(-12)=-2+12=10 \\ \text{ the radius is half the diameter }\]

Answer 26

oh got it, thanks so much for your help and patience