<< Chapter < Page Chapter >> Page >

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.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

24 + ( −12 ) + 6 + ( −3 ) + ...

The sum of the infinite series is defined.

Got questions? Get instant answers now!

k = 1 15 ( 0.3 ) k

The sum of the infinite series is defined.

Got questions? Get instant answers now!

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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask