MATH SOLVE

3 months ago

Q:
# 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? 5+(−10)z+20z2+(−40)z3+⋯

Accepted Solution

A:

[tex]5\times(-2)^0=5[/tex]

[tex]5\times(-2)^1=-10[/tex]

[tex]5\times(-2)^2=20[/tex]

[tex]5\times(-2)^3=-40[/tex]

...

There's enough of a pattern here to discern the series to be

[tex]\displaystyle\sum_{k=0}^\infty 5(-2z)^k[/tex]

which converges as long as [tex]|-2z|=2|z|<1[/tex], or [tex]|z|<\dfrac12[/tex]. Under this condition, the series would converge to the function

[tex]f(z)=\dfrac5{1+2z}[/tex]

[tex]5\times(-2)^1=-10[/tex]

[tex]5\times(-2)^2=20[/tex]

[tex]5\times(-2)^3=-40[/tex]

...

There's enough of a pattern here to discern the series to be

[tex]\displaystyle\sum_{k=0}^\infty 5(-2z)^k[/tex]

which converges as long as [tex]|-2z|=2|z|<1[/tex], or [tex]|z|<\dfrac12[/tex]. Under this condition, the series would converge to the function

[tex]f(z)=\dfrac5{1+2z}[/tex]