simplifying a summation for a time series

I was curious how to simplify this summation that is giving me a bit of trouble

$Y_t$=$-\sum_{j=1}^\infty (\frac{1}{3})^j*e_{t+j}$

So i get that this is a geometric sum that converges and I would was thinking I would do the following

Sum = -$\frac{\frac{1}{3}*e_{t+1}}{1-\frac{1}{3}}=-\frac{1}{2}*e_{t+1}$

but the index on $e_{t+j}$ is causing me some problems. Do I need to take the $e_{t+j}$ into account in the denominator when I subtract the ratio?

Any help would be appreciated.

Answer

You can approximate the summation as you have tried:

$$\text{Sum}\approx-\frac12e_{t+1}$$

If you wanted to other values of $e_{t+j}$ into consideration,

$$\text{Sum}\approx-\frac12\operatorname{avg}(\{e_{t+j}\}_{j=1}^\infty)$$

Or, you could use the old fashioned method:

$$\text{Sum}=-\frac13e_{t+1}-\frac19e_{t+2}-\frac1{27}e_{t+3}-\dots$$

And just cut the sum short once you feel accurate enough.