Determine if the following series converges or diverges. If...

Determine if the following series converges or diverges. If the series converges give its value. a) sum_{n=2}^{infty} frac{2^{n+1} - 5}{4^n} b) sum_{n=1}^{infty} frac{15}{n^2 + 7n + 12}

Transcript

00:01 Now let me to find the convergence of this series now let's compare it with summation one over two wrist power and as in goes to infinity from two starting from two now an is this and bn is this so let's compare a n over bn or bn comparison test as in goes to infinity we have 2x power n plus 1 minus 5 or 4 power n times 2st power n so this becomes we can 2st power n plus 1 minus 5 over 2 is power n now we can divide the numerator denominator by 2 is power n so this becomes 2 minus 5 over 2 s power n over 1 and as n goes to infinity this goes to 2 therefore if bn converges we cannot use this test to conclude a in converges as well actually we can since since bn converges therefore a n must converge as well since we are getting a constant value as n goes to infinity now we can break this series into two series that would be 2x power n plus 1 divided by 4 s power n minus summation 5 over 4 is power n and and n goes from 2 to infinity in both cases so each of these two is a geometric series this first one can be simplified to 2 rous power n plus 1 minus 2 n that would be 2 2 power 1 minus n or in other words 2 divide by 2 plus n then goes from 2 to infinity and the second one can be written in the same way now we can use sum to infinity of geometric series and that is equals to a over 1 minus r we have for the first series a would be 2 divided by 4 that is half over 1 minus and common ratio is also half minus the first term for the second series would be 5 over 16 and divide by one minus ratio that is 1 over 4 so this becomes half one minus half is a half this is one and 5 or 16 divided by 3 or 4 that is 4 over 3 so this becomes 5 over 12 1 minus 5 over 12 this is equals to 7 over 12 for the first series for the second series we have summation 15 over n squared the 7 n plus 12 and then goes from 1 to infinity now this can be broken into partial fractions as a n squared plus 7 and plus 2 can be factorized to n plus 3 times n plus 4 breaking this into partial fractions as a divided by n plus 3 plus b divided by n plus 4 we will then multiply both sides of equation by n plus 3 times n plus 4 and you get a times n plus 4 plus 4 plus b times n plus 3 is equals 15 expanding the expressions we get a plus b times n is equals plus 4 a plus 3b is equals to 15 now we know there is no end term on the left hand side so a plus b must be equals zero and we have 4 a plus 3b is equals to 15 now b must be equals to negative of a.

06:26 Substituting this, we get 4a minus 3a, a is equals to 15, and we get b is equal to negative 15...

Question

Determine if the following series converges or diverges. If the series converges give its value. a) sum_{n=2}^{infty} frac{2^{n+1} - 5}{4^n} b) sum_{n=1}^{infty} frac{15}{n^2 + 7n + 12}

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