Question

Use the properties of geometric series to find the sum of the series. for what values of the variable does the series converge to this sum? 7−14z+28z2−56z3+⋯

Answer 1

Answer:

Z=1

Step-by-step explanation:

This is a geometric progression

7-14Z+28Z^2-56Z^3

a=7

r=2

Sn=a(r^n-1)/(r-1)

Sn=7(2^n-1)/1

Sn=7(2^n-1)

Sinfinity=a/(1-r)

Sinfinity=7

7(2^n-1)=7

2^n-1=1

2^n=2

n=1

Answer 2

Answer:

- The sum of the series is given as

7/(1 + 7z)

- The series converges for (-1/7, 1/7)

Step-by-step explanation:

Given the geometric series:

7 - 14z + 28z² - 56z³ + ...

The common ratio r is given as

-14z/7 or 28z²/(-14z) or -56z³/(28z²) and so on

r = -7z

The sum of the series is given as

S = a/(1 - r)

Where a is the first term in the series.

S = 7/(1 -(-7z))

= 7/(1 + 7z)

The series converges for |-7z| < 1

That is for -1/7 < z < 1/7