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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} $

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error $\approx 0.00002$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Campbell University

Baylor University

Idaho State University

Boston College

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

03:30

Use the sum of the first 1…

03:05

03:03

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.

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