9. For what constants λand m does the infinite series (1^3...

9. For what constants \lambda and $m$ does the infinite series \\ $\frac{1^3 \lambda}{1!} e^{-m} + \frac{2^3 \lambda^2}{2!} e^{-2m} + \frac{3^3 \lambda^3}{3!} e^{-3m} + \frac{4^3 \lambda^4}{4!} e^{-4m} + \dots$ \\ converge, and to what sum?

Transcript

00:01 Hi, so today we're going to be looking at telescoping series and binding the nth term and then evaluating the limit of that nth term.

00:12 So right now we have this equation and we're just going to rewrite this.

00:16 So we're going to rewrite this as 1 over 4k plus 3 times 4k minus.

00:32 Minus 1.

00:35 And now we're going to take this equation down and we're going to work with.

00:45 Now using method learned with partial fractions, we're just on this side going to be writing out, sorry, 1 over 4k plus 3 minus 1 over 4k plus 3 minus 1 over 4.

01:12 K minus 1 and now from here what we're going to do is we're going to multiply um up here by 4k minus 1 minus 4k uh plus 3 so we're going to end up with this having a 4k minus 1 up here um minus 4 k minus 3 with 4k plus 3 times 4k minus 1.

02:05 So we can now simplify and we should get rid of those values.

02:12 And we should be left with negative 4 over 4k plus 3 times 4k minus.

02:27 And now from here, what we're going to do is we're going to bring down this equation to here.

02:39 So we're going to say this is equal to 1 over 4k plus 3 minus 1 over 4k minus 1.

02:50 And we're going to divide both sides by negative 1 fourth.

03:02 And divide both sides by negative 4.

03:09 And now we're going to be dealing with this equation.

03:11 This equation is going to then be equal to 1 4th times 1 over 4k minus 1 over 4k plus 3.

03:31 So that's going to be the equation that we have there.

03:34 From here, what we're going to do, though, is we need to start plugging some numbers in and then solving to find our nth term.

03:46 So we're going to have that s n is going to be equal to, we can take out our one -fourth here, and we're going to have this sigma of n, and this is going to be when k is equal of this equation.

04:07 Okay.

04:08 And then now from here we're going to be able to start plugging in our values.

04:14 So we're going to say bring down our one -fourth.

04:17 Just got to remember that it's there.

04:19 So we're going to plugging in for k -equal to zero...

Question

9. For what constants \lambda and $m$ does the infinite series \\ $\frac{1^3 \lambda}{1!} e^{-m} + \frac{2^3 \lambda^2}{2!} e^{-2m} + \frac{3^3 \lambda^3}{3!} e^{-3m} + \frac{4^3 \lambda^4}{4!} e^{-4m} + \dots$ \\ converge, and to what sum?

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