Denote the [tex]n[/tex]th partial sum of the series by
[tex]S_n=5-\dfrac53+\dfrac59-\dfrac5{27}+\cdots+5\left(-\dfrac13\right)^{n-1}+5\left(-\dfrac13\right)^n[/tex]
Multiplying both sides by [tex]-\dfrac13[/tex], we have
[tex]-\dfrac13S_n=-\dfrac53+\dfrac59-\dfrac5{27}+\cdots+5\left(-\dfrac13\right)^n+5\left(-\dfrac13\right)^{n+1}[/tex]
Subtracting the second equation from the first gives
[tex]S_n-\left(-\dfrac13\right)S_n=5-5\left(-\dfrac13\right)^{n+1}[/tex]
[tex]\dfrac43S_n=5-5\left(-\dfrac13\right)^{n+1}[/tex]
[tex]S_n=\dfrac{15}4\left(1-\left(-\dfrac13\right)^{n+1}\right)[/tex]
As [tex]n\to\infty[/tex], the geometric term approaches 0, leaving you with
[tex]S_\infty=\dfrac{15}4[/tex]
