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Chapter 9: Problem 70

Identify the two series that are the same. (a) \(\sum_{n=4}^{\infty} n\left(\frac{3}{4}\right)^{n}\) (b) \(\sum_{n=0}^{\infty}(n+1)\left(\frac{3}{4}\right)^{n}\) (c) \(\sum_{n=1}^{\infty} n\left(\frac{3}{4}\right)^{n-1}\)

Short Answer

Expert verified
The two series that are the same are series (a) and series (c), when (c) is rewritten to start from \(n=4\).

Step by step solution

01

Compare Series (a) and (b)

First, we compare series (a) and (b). Series (a) starts at \(n=4\) while series (b) starts at \(n=0\). Since the series are different in terms of their starting index, they can't be the same.
02

Compare Series (a) and (c)

Now let's compare series (a) and (c). In series (c), the index starts from \(n=1\), while in series (a), from \(n=4\). We write the first few terms for series (c) as \(1 \cdot \left(\frac{3}{4}\right)^0, 2 \cdot \left(\frac{3}{4}\right)^1, 3 \cdot \left(\frac{3}{4}\right)^2, 4 \cdot \left(\frac{3}{4}\right)^3, \ldots\). We can see that starting from the fourth term, it matches with series (a). It means we can rewrite series (c) to start from \(n=4\) by eliminating the first three terms and it would become the same as series (a). Therefore, series (a) is the same as series (c).
03

End Comparison

We have found that series (a) and (c) are the same. The beginning of our series and the constants with which we multiply our variable have changed, but they still add up to the same solution. There is no need to check series (b) because it was mentioned that there are only two series that are the same.

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Most popular questions from this chapter

Finding a Maclaurin Series In Exercises \(41-44\) , find the Maclaurin series for the function. (See Examples 7 and \(8 . )\) $$ f(x)=x \sin x $$

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