Simplifying an Infinite Sum

[eastbay_cutie]

Is this valid?

P = {Sum from j=0 to infinity of} ((1 + i - L)^j)/((1 + i)^j)


P = [(1 + i - L)/(1 + i)]

 

[pka]

P = {Sum from j=0 to infinity of} ((1 + i - L)^j)/((1 + i)^j)

Do you know that \(\displaystyle \left| {1 - \frac{L}{{1 + i}}} \right| < 1~?\)

 

[eastbay_cutie]

sorry, L and i are between 0 and 1.

 

[pka]

P = {Sum from j=0 to infinity of} ((1 + i - L)^j)/((1 + i)^j)

If \(\displaystyle |r|<1\) then \(\displaystyle \displaystyle\sum\limits_{j = 0}^\infty {r^j } = \frac{1}{{1 - r}}\)

Notice \(\displaystyle \dfrac{{\left( {1 + i - L} \right)^j }}{{\left( {1 + i} \right)^j }} = \left( {1 - \dfrac{L}{{1 + i}}} \right)^j \)