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This series can also be written in summation notation as k = 1 2 k , where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges .

Determining whether the sum of an infinite geometric series is defined

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

1 + 0.2 + 0.04 + 0.008 + 0.0016 + ...

The common ratio r  = 0 .2 . As n gets very large, the values of r n get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with 1 < r < 1 approach 0; the sum of a geometric series is defined when 1 < r < 1.

Determining whether the sum of an infinite geometric series is defined

The sum of an infinite series is defined if the series is geometric and 1 < r < 1.

Given the first several terms of an infinite series, determine if the sum of the series exists.

  1. Find the ratio of the second term to the first term.
  2. Find the ratio of the third term to the second term.
  3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
  4. If a common ratio, r , was found in step 3, check to see if 1 < r < 1 . If so, the sum is defined. If not, the sum is not defined.

Determining whether the sum of an infinite series is defined

Determine whether the sum of each infinite series is defined.

  1. 12 + 8 + 4 + 
  2. 3 4 + 1 2 + 1 3 + ...
  3. k = 1 27 ( 1 3 ) k
  4. k = 1 5 k
  1. The ratio of the second term to the first is 2 3 , which is not the same as the ratio of the third term to the second, 1 2 . The series is not geometric.
  2. The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of 2 3 . The sum of the infinite series is defined.

  3. The given formula is exponential with a base of 1 3 ; the series is geometric with a common ratio of 1 3 . The sum of the infinite series is defined.
  4. The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.
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Determine whether the sum of the infinite series is defined.

1 3 + 1 2 + 3 4 + 9 8 + ...

The sum is defined. It is geometric.

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24 + ( −12 ) + 6 + ( −3 ) + ...

The sum of the infinite series is defined.

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k = 1 15 ( 0.3 ) k

The sum of the infinite series is defined.

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Finding sums of infinite series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first n terms of a geometric series.

S n = a 1 ( 1 r n ) 1 r

We will examine an infinite series with r = 1 2 . What happens to r n as n increases?

( 1 2 ) 2 = 1 4 ( 1 2 ) 3 = 1 8 ( 1 2 ) 4 = 1 16

The value of r n decreases rapidly. What happens for greater values of n ?

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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