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Find the radius of convergence, then determine the interval of convergence

galben [10]

The radius of convergence R is 1 and the interval of convergence is (-3, -1) for the given power series. This can be obtained by using ratio test.

<h3>Find the radius of convergence R and the interval of convergence:</h3>

Ratio test is the test that is used to find the convergence of the given power series.

First aₙ is noted and then aₙ₊₁ is noted.

For ∑ aₙ, aₙ and aₙ₊₁ is noted.

$\lim_{n \to \infty} |\frac{a_{n+1}}{a_{n} }|$ = β

If β < 1, then the series converges

If β > 1, then the series diverges

If β = 1, then the series inconclusive

Here $a_{k}$ = $\frac{(x+2)^{k}}{\sqrt{k} }$ and $a_{k+1}$ = $\frac{(x+2)^{k+1}}{\sqrt{k+1} }$

Now limit is taken,

$\lim_{n \to \infty} |\frac{a_{n+1}}{a_{n} }|$ = $\lim_{n \to \infty} |\frac{(x+2)^{k+1} }{\sqrt{k+1} }/\frac{(x+2)^{k} }{\sqrt{k} }|$

= $\lim_{n \to \infty} |\frac{(x+2)^{k+1} }{\sqrt{k+1} }\frac{\sqrt{k} }{(x+2)^{k}}|$

= $\lim_{n \to \infty} |{(x+2) } }{\sqrt{\frac{k}{k+1} } }}|$

= $|{x+2 }|\lim_{n \to \infty}}{\sqrt{\frac{k}{k+1} } }}$

= $|{x+2 }|$ < 1

- 1 < ${x+2 }$ < 1

- 1 - 2 < x < 1 - 2

- 3 < x < - 1

We get that,

interval of convergence = (-3, -1)

radius of convergence R = 1

Hence the radius of convergence R is 1 and the interval of convergence is (-3, -1) for the given power series.

Learn more about radius of convergence here:

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5 0

1 year ago

Read 2 more answers

which is not a result of multiplying the height of a rectangular prism by a scale factor of 1/2 and I need work to

chubhunter [2.5K]

C is the answer

for example

assume that the height of a rectangular prism is 2, width is 1 and length measures 3

the volume will be 6
(2*1*3)

if we multiply height with the scale factor of 1/2
it becomes 1

so the volume will be 3
(1*1*3)

this situation goes for other examples, too

good luck

5 0

3 years ago

this morning, jose's car had 16.01 gallons of fuel. now, 1.5 gallons are left. how much fuel did jose use ?

Gekata [30.6K]

Answer:

Jose used 14.51 gallons of fuel

8 0

3 years ago

Complete the table.<br><br> f(x)=2x+4<br><br> 0 _<br> 1 _ <br> 2 _<br> 3 _

Westkost [7]

Answer:

0,4

1,6

2,8

3,10

Step-by-step explanation:

if x=0, 2*0+4=0+4=4

if x=1, 2*1+4=2+4=6

if x=2, 2*2+4=4+4=8

if x=3, 2*3+4=6+4=10

3 0

3 years ago

Use the digits 0-9, find out how many 4 digit numbers can be configured based on the stated conditions:

Alexeev081 [22]

A) On the first place we can put 9 digits, on the second 9, on the third place 8 and on the last place 7 ( because no digits can be repeated ):
9 · 9 · 8 · 7 =4536 numbers
b) There are 5 odd digits can be repeated. So:
5 · 10 · 10 · 5 = 2500 numbers
c) On the first place we can put:5,6,7,8,9 (5 digits) On the last place is just one (0), because the number must be divisible by 10:
5 · 10 · 10 · 1 = 500 numbers
d) on the first place we can put 2 digits ( 1, or 2 - there are no 4-digit numbers starting with 0). The last digit must be even, but the order can be:
X - odd - odd - even 2 · 5 · 4 · 5 = 200
X - odd - even - even 2 · 5 · 5 · 4 = 200
X - even - odd - even 2 · 5 · 5 · 4 = 200
X - even - even - even 2 · 5 · 4 · 3 = 120
Finally: 200+ 200 + 200 + 120 = 720 numbers

3 0

3 years ago