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Tanya [424]
2 years ago
14

Let f(x) = [infinity] xn n2 n = 1. find the intervals of convergence for f. (enter your answers using interval notation. ) find

the intervals of convergence for f '. find the intervals of convergence for f ''
Mathematics
1 answer:
inna [77]2 years ago
6 0

Best guess for the function is

\displaystyle f(x) = \sum_{n=1}^\infty \frac{x^n}{n^2}

By the ratio test, the series converges for

\displaystyle \lim_{n\to\infty} \left|\frac{x^{n+1}}{(n+1)^2} \cdot \frac{n^2}{x^n}\right| = |x| \lim_{n\to\infty} \frac{n^2}{(n+1)^2} = |x| < 1

When x=1, f(x) is a convergent p-series.

When x=-1, f(x) is a convergent alternating series.

So, the interval of convergence for f(x) is the <em>closed</em> interval \boxed{-1 \le x \le 1}.

The derivative of f is the series

\displaystyle f'(x) = \sum_{n=1}^\infty \frac{nx^{n-1}}{n^2} = \frac1x \sum_{n=1}^\infty \frac{x^n}n

which also converges for |x| by the ratio test:

\displaystyle \lim_{n\to\infty} \left|\frac{x^{n+1}}{n+1} \cdot \frac n{x^n}\right| = |x| \lim_{n\to\infty} \frac{n}{n+1} = |x| < 1

When x=1, f'(x) becomes the divergent harmonic series.

When x=-1, f'(x) is a convergent alternating series.

The interval of convergence for f'(x) is then the <em>closed-open</em> interval \boxed{-1 \le x < 1}.

Differentiating f once more gives the series

\displaystyle f''(x) = \sum_{n=1}^\infty \frac{n(n-1)x^{n-2}}{n^2} = \frac1{x^2} \sum_{n=1}^\infty \frac{(n-1)x^n}{n} = \frac1{x^2} \left(\sum_{n=1}^\infty x^n - \sum_{n=1}^\infty \frac{x^n}n\right)

The first series is geometric and converges for |x|, endpoints not included.

The second series is f'(x), which we know converges for -1\le x.

Putting these intervals together, we see that f''(x) converges only on the <em>open</em> interval \boxed{-1 < x < 1}.

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For what real value of $v$ is $\frac{-21-\sqrt{301}}{10}$ a root of $5x^2+21x+v$?
Degger [83]

Answer:

v = 7

is the value for which

x = (-21 - √301)/10

is a solution to the quadratic equation

5x² + 21x + v = 0

Step-by-step explanation:

Given that

x = (-21 - √301)/10 .....................(1)

is a root of the quadratic equation

5x² + 21x + v = 0 ........................(2)

We want to find the value of v foe which the equation is true.

Consider the quadratic formula

x = [-b ± √(b² - 4av)]/2a ..................(3)

Comparing (3) with (2), notice that

b = 21

2a = 10

=> a = 10/2 = 5

and

b² - 4av = 301

=> 21² - 4(5)v = 301

-20v = 301 - 441

-20v = -140

v = -140/(-20)

v = 7

That is a = 5, b = 21, and v = 7

The equation is then

5x² + 21x + 7 = 0

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3 years ago
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