MATH SOLVE

6 months ago

Q:
# Find the sum of the series. 1 + z 3 + z2 9 + z3 27 + $$ correct: your answer is correct. for what values of the variable does the series converge to this sum? (enter your answer using interval notation.)

Accepted Solution

A:

[tex]1+\dfrac z3+\dfrac{z^2}9+\dfrac{z^3}{27}+\cdots=\displaystyle\sum_{n=0}^\infty\left(\frac z3\right)^n[/tex]

which converges for [tex]\left|\dfrac z3\right|<1[/tex] or [tex]|z|<3[/tex] to [tex]\dfrac1{1-\frac z3}=\dfrac3{3-z}[/tex].

which converges for [tex]\left|\dfrac z3\right|<1[/tex] or [tex]|z|<3[/tex] to [tex]\dfrac1{1-\frac z3}=\dfrac3{3-z}[/tex].