Answer
Limit: $1$
Work Step by Step
We are given the sequence:
$a_n=1+\left(-\dfrac{1}{2}\right)^n$.
Determine the first 10 terms:
$a_1=1+\left(-\dfrac{1}{2}\right)^1=0.5$
$a_2=1+\left(-\dfrac{1}{2}\right)^2=1.25$
$a_3=1+\left(-\dfrac{1}{2}\right)^3=0.875$
$a_4=1+\left(-\dfrac{1}{2}\right)^4=1.0625$
$a_5=1+\left(-\dfrac{1}{2}\right)^5=0.968755$
$a_6=1+\left(-\dfrac{1}{2}\right)^6=1.015625$
$a_7=1+\left(-\dfrac{1}{2}\right)^7=0.9921875$
$a_8=1+\left(-\dfrac{1}{2}\right)^8=1.0039063$
$a_9=1+\left(-\dfrac{1}{2}\right)^9=0.99804688$
$a_{10}=1+\left(-\dfrac{1}{2}\right)^{10}=1.0009766$
The sequence appears to have a limit: 1.
Calculate the limit:
$\displaystyle{\lim_{n \to \infty}} \left[1+\left(-\dfrac{1}{2}\right)^n\right]=1+\displaystyle{\lim_{n \to \infty}}\left(-\dfrac{1}{2}\right)^n$
$=1+0=1$
Plot the first 10 terms of the sequence:
