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Artist 52 [7]
3 years ago
12

Find the radius of convergence, r, of the series. ? n2xn 7 · 14 · 21 · ? · (7n) n = 1

Mathematics
1 answer:
defon3 years ago
3 0
I'm guessing the series is supposed to be

\displaystyle\sum_{n=1}^\infty\frac{n^2x^n}{7\cdot14\cdot21\cdot\cdots\cdot(7n)}

By the ratio test, the series converges if the following limit is less than 1.

\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2x^{n+1}}{7\cdot14\cdot21\cdot\cdots\cdot(7n)\cdot(7(n+1))}}{\frac{n^2x^n}{7\cdot14\cdot21\cdot\cdots\cdot(7n)}}\right|

The first n terms in the numerator's denominator cancel with the denominator's denominator:

\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2x^{n+1}}{7(n+1)}}{n^2x^n}\right|

|x^n| also cancels out and the remaining factor of |x| can be pulled out of the limit (as it doesn't depend on n).

\displaystyle|x|\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2}{7(n+1)}}{n^2}\right|=|x|\lim_{n\to\infty}\frac{|n+1|}{7n^2}=0

which means the series converges everywhere (independently of x), and so the radius of convergence is infinite.
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Answer:

It takes the frog 7 hours to get out of the well.

Step-by-step explanation:

We know that the well is 11 feet deep and the frog can climb 3 feet per hour.

Each time it climbs, it rests for an hour, and decreases its height by 1 foot.

So, if the frog reaches 3 feet in the first hour, then in the next hour it is 2 feet.

Lets calculate with the same pattern:

1st hour: 3 feet

2nd hour: 3-1=2 feet

3rd hour: 2+3=5 feet

4th hour: 5-1=4 feet

5th hour: 4 +3= 7 feet

6th hour: 7-1=6 feet

5th hour: 6+3=9 feet

6th hour: 9-1=8 feet

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Therefore, it takes the frog 7 hours to get out of the well.

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Answer:

true!

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