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

I Need Help Fast Please

lbvjy [14]

C. 130,90,130.

#01234, the tenths number stays the same
#56789, the tenths number go up 1 number

4 0

3 years ago

15. What polynomial must be added to .x^2-2x+ 6 so that<br> the sum is 3x^2+ 7x ?

jeyben [28]

Answer:

$2x^2+9x-6$

Step-by-step explanation:

Just subtract the two because x + y = z and z-x = y

$3x^2+7x-(x^2-2x+6) = 3x^2+7x-x^2+2x-6\\\\=2x^2+9x-6$

You can test this by adding this expression back to the original.

$x^2-2x+6+(2x^2+9x-6) = 3x^2+7x$

8 0

3 years ago

How do you solve this using substitution <br> -8x-y=-30<br> Y=12x-50

never [62]

Step-by-step explanation:

substitude the y value into y in the first equation

-8-(12x-50)=30

-8-12x+50=30

you see it?

then solve

-8+50-30=12x

you got the value = 12x

then you divide with 12

finally you get x

7 0

3 years ago