Find the sum of the infinite series

Question

there's an infinite sign over the E and j-1 on the bottom then beside it it is 4*0.5^j-1

 

please help I know that may be very confusing.

 

http://152.46.13.240/MoodleContent/Math3/Module6/Lesson%207/eq_ecd1db.gif

 

The link above will show you the equation.

 

Answer 1

Here's how you do it.

 

 

First rewrite ∑4•(0.5)^j-1 as 4•∑(0.5)^j-1 

 

Then ∑(0.5)^j-1 is an infinite geometric series with j starting at 1.

 

The formula for the sum of an infinite geometric series is a/(1 - r), a is the first term and r is the common ratio.

 

Substituting j =1 into (0.5)^j-1 we get (0.5)^1-1 = (0.5)⁰ = 1. The common ratio is r = 0.5.

 

So the sum of Σ(0.5)^j-1 = 1/(1-0.5) = 2

 

So then Σ4•(0.5)^j-1 = 4•Σ(0.5)^j-1 = 4•2 = 8

 

And that's it! I hope this helped a little.