1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Mekhanik [1.2K]
3 years ago
15

Which of the following geometric series converges?

Mathematics
1 answer:
Artist 52 [7]3 years ago
8 0

All three series converge, so the answer is D.

The common ratios for each sequence are (I) -1/9, (II) -1/10, and (III) -1/3.

Consider a geometric sequence with the first term <em>a</em> and common ratio |<em>r</em>| < 1. Then the <em>n</em>-th partial sum (the sum of the first <em>n</em> terms) of the sequence is

S_n=a+ar+ar^2+\cdots+ar^{n-2}+ar^{n-1}

Multiply both sides by <em>r</em> :

rS_n=ar+ar^2+ar^3+\cdots+ar^{n-1}+ar^n

Subtract the latter sum from the first, which eliminates all but the first and last terms:

S_n-rS_n=a-ar^n

Solve for S_n:

(1-r)S_n=a(1-r^n)\implies S_n=\dfrac a{1-r}-\dfrac{ar^n}{1-r}

Then as gets arbitrarily large, the term r^n will converge to 0, leaving us with

S=\displaystyle\lim_{n\to\infty}S_n=\frac a{1-r}

So the given series converge to

(I) -243/(1 + 1/9) = -2187/10

(II) -1.1/(1 + 1/10) = -1

(III) 27/(1 + 1/3) = 18

You might be interested in
Question 1 of 25
aleksklad [387]
SAS i believe :) sorry if im wrong!
8 0
3 years ago
The line 3x+4y=24 cuts the x axis at point A and y axis at point B<br><br> Find the lenght if AB
sweet [91]
Length A is (8,0) and the length of B is (0,6)
4 0
3 years ago
Pela has 5 3/4 feet of string to make necklaces. She wants to make 4 necklaces that are the same length. How long should each ne
antiseptic1488 [7]
5.75/4 = 1.4375 feet a necklace
4 0
3 years ago
Read 2 more answers
Question in pic! help asap!! will give brainliest
Viktor [21]

Answer:

a6b40c70

Step-by-step explanation:

a6b40c70

6 0
2 years ago
Read 2 more answers
X^2+25x-156=0 ayúdenme
Sonbull [250]
X= -25/2 + 1/2 _/1249
3 0
3 years ago
Other questions:
  • 18 of the last 30 cars that passed you were black what is the probability the next car will be black
    11·1 answer
  • Find the? inverse, if it? exists, for the given matrix.<br><br> [4 3]<br><br> [3 6]
    7·1 answer
  • A population y of coyotes in a national park triples every 20 years. The function y = 15(3)^x
    11·1 answer
  • Find the area of the shape shown below.
    5·1 answer
  • Determine the difference: 3.2 × 1010 – 1.1 × 1011 Write your answer in scientific notation.
    14·1 answer
  • What is the height of the shape formed by the sectors shown below?
    14·2 answers
  • I don’t get it I need a little help
    11·2 answers
  • How i can answer this question, NO LINKS, if you answer correctly i will give u brainliest!
    5·1 answer
  • if the value of c is negative, you would need zero pairs to model the factorization of the polynomial. the x-tiles on the board
    7·1 answer
  • Finding the sums
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!