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JH
Numerade Educator

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Problem 36 Medium Difficulty

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{3^n + 4^n} $

Answer

error $\approx 0.00002$

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Video Transcript

let's use the some of the first ten terms to approximate the Siri's. So here's the Siri's on the left that's approximately as ten. And then let's find let's approximate as ten. Actually, here's the exact answer, or someone Wolfram Alpha you could see on the left over here. The sum from one to ten. That's the sum that we want. Here's the exactly answer. If you want, it is a large fraction, too right out. But if you'LL settle for the that's more representation, then you Khun do point one nine eight. So that's our approximation to the actual son. And then I will find the year from using ten terms. So there is less than or equal to our ten. That's the remainder when using ten terms to approximate the song. Now for this theories here. If we have to show whether this thing converges, we could do a comparison, say, with one over three, then and then for this. Siri's here. This convergence is geometric, but if we've let's to know by FX dysfunction right here, then you can show that African defense. Let's see if it's positive. That's true continuous. This is true as well. Even differential and it's decreasing. It's clear to see that is decreasing. Every time X gets bigger on ly, the denominator gets larger, so the fraction is the hole gets smaller, so depressing. So here we can use the upper bound that you would get from the integral test. So this in apologies justified on Paige three seventy Excuse me, seventy. So this is just the integral from ten to infinity. So this is the upper bound for the air when using the integral test. So let's go and evaluate that integral. So that ends up being That's an infinity of there. So this we can find the exact answer by just plugging in the endpoints. Or we can just approximate this part of the calculator. So that's a decimal. The little needle here zero point zero zero zero zero two. So this is the upper bound for the air, and that's my final answer.