Algebra or calculus?

If $y = e^{x/3}$ , then $y' = \frac{1}{3}e^{x/3}, y'' = \frac{1}{9}e^{x/3}, y''' = \frac{1}{27}e^{x/3}, .... , y^{(n)} = \frac{1}{3^n}e^{x/3}$ . Hence, the infinite series computes as:

$\Sigma_{n=1}^{\infty} \frac{e^{x/3}}{3^n}$ ;

or $e^{x/3} \cdot \Sigma_{n=1}^{\infty} (\frac{1}{3})^n$ ;

or $e^{x/3} \cdot \frac{1/3}{1 - 1/3} = \frac{1}{2}e^{x/3}.$

At $x=3$ , we finally obtain $\boxed{\frac{e}{2} = 1.36}$ (to 2 decimal places).

It is of the form y' +y''+y'''+y''''......... ==> e^(x/3). *( 1/3 + 1/9 + 1/27 + 1/81......) Y/2....(y=e^x/3)..... When x=3, the and is e/2=== 1.3591....