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Chapter 14: Problem 40

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$\frac{1}{9}+\frac{2}{9^{2}}+\frac{3}{9^{3}}+\dots+\frac{n}{9^{n}}$$

Short Answer

Expert verified
The sum expressed in sigma notation is \(\sum_{i=1}^{n}\frac{i}{9^{i}}\).

Step by step solution

01

Recognize The Pattern

First we need to identify the pattern. For each term, it appears that the numerator is equal to the term number and the denominator is 9 raised to the term number. The terms of the series are in the form \(\frac{i}{9^{i}}\) where i goes from 1 to n.
02

Writing in summation notation

Now, using the pattern identified, the sum can be written using sigma (危) notation which often used to denote sum. The variable 'i' will be the index of summation which starts from the lower limit 1 and goes up to an upper limit 'n'. Using sigma notation our sum becomes \(\sum_{i=1}^{n}\frac{i}{9^{i}}\)

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Most popular questions from this chapter

A deposit of 6000 dollars is made in an account that earns \(6 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by the sequence $$a_{n}=6000\left(1+\frac{0.06}{4}\right)^{n}, \quad n=1,2,3, \ldots$$ Find the balance in the account after five years. Round to the nearest cent.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I modeled California's population growth with a geometric sequence, so my model is an exponential function whose domain is the set of natural numbers.

What is a geometric sequence? Give an example with your explanation.

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=3 \cdot 5^{n}\). Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$5,7,9,11, \dots$$
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