SOLUTION: Consider the infinite geometric series n=1 up to infinitey then the equation is -4(1/3)^n-1 a. write the first four terms of the series b. does the series diverge or converg

Question 83773: Consider the infinite geometric series
n=1 up to infinitey then the equation is -4(1/3)^n-1

a. write the first four terms of the series
b. does the series diverge or converge?
c. If the series has a sum, find the sum
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
In order to find the terms of the series we need to find the terms of the sequence first.

If we start at n=1, our first number of the sequence is

If we let n=2, our second number of the sequence is

If we let n=3, our third number of the sequence is

If we let n=4, our fourth number of the sequence is

If we let n=5, our fifth number of the sequence is

If we let n=6, our sixth number of the sequence is

and so on...

a. write the first four terms of the series
Now that we've generated the sequence of numbers, lets generate the series

So now lets add up the individual terms of the sequence we found earlier
Sum of the first 2 terms of the sequence is

Sum of the first 3 terms of the sequence is

Sum of the first 4 terms of the sequence is

Sum of the first 5 terms of the sequence is

So the first four terms are

,,,

b. does the series diverge or converge?
If we let the series go on long enough, the series will converge. It's hard to see with the fractions, but the decimal values are clear enough. The series goes
-5.3333333333333 , -5.77777777777778 , -5.92592592592593 , -5.97530864197531...
where the 11th term is -5.99989838947315 and the 32th term is -5.99999999999999. So the series converges to -6.

c. If the series has a sum, find the sum

To find the sum of an infinite series, use this formula

where S is the sum, a is the first term (in this case -4) and r is the ratio (in this case )

plug in a=-4 and

Make 1 into an equivalent fraction with a denominator of 3

Combine the fractions in the denominator

Multiply the fractions by multiplying the first fraction by the reciprocal of the second

Multiply

Reduce

So the infinite sum converges to -6
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