Exercise 201 Page 558

a We are given an infinite series.

∑_(n=0)^(∞) 4(- 2/3)^n We want to determine whether the series is geometric or arithmetic. Recall the general formulas for the n^\text{th} term of arithmetic and geometric series.

Arithmetic	Geometric
t(n) = t(0) + n* d	t(n) = t(0)* r^n

In an arithmetic series, the constant common difference is added to each term to find the next one. In a geometric series, a term is multiplied by value to find the next term. Let's take a look at the n^\text{th} term of our series. t(n) = 4 * (- 2/3 )^n The n^\text{th} term is given by t(n) = 4(- 23)^n, which follows the general formula for the n^\text{th} term of the geometric series. This means that our series is geometric!

b Now we want to determine whether the given series has a finite sum. A series converges when the absolute values of the terms get smaller and smaller over time — eventually the value of each term is so small that it does not actually affect the sum by much anymore. Consider the given series.

t(n) = 4 * (- 2/3 )^n In Part A, we established it is a geometric. series Let's identify the common ratio r. t(n) = 4 * ( - 2/3 )^n ⇕ a = 4 r = - 2/3 We can see that the common ratio is - 23. To find out if the series converges, we will calculate the absolute value of this ratio. Because the common ratio is the multiplicative factor between the terms, if its absolute value is less than 1, the absolute value of the terms will decrease over time. |r|=|- 2/3|=2/3 The absolute value of the common ratio is less than 1, so the series does converge to a value.

c Now we want to find the sum of the series. Since the absolute value of the common ratio is less than 1, the series converges to a value, meaning that we can calculate its sum. Recall the formula for the sum of an infinite geometric series.

∑^(∞)_(n=0) ar^n = a/1-r In this formula, r is the common ratio and a is the first term in the series. In Part B we found that a = 4 and r = - 23. Let's substitute these values into our formula to evaluate the sum of the series!

a/1-r

SubstituteII

r= - 2/3, a= 4

4/1-( - 23)

SubNeg

a-(- b)=a+b

4/1+ 23

OneToFrac

Rewrite 1 as 3/3

4/33+ 23

AddFrac

Add fractions