AoPS Wiki talk:Problem of the Day/August 1, 2011
We can split this summation, as shown: .
Now we must simply find each of the smaller sums, and add them.
We can use the formula for the sum of an infinite geometric series ( where is the first term and is the common ratio). and since each term is getting multiplied by to receive the next term. Therefore, this sum is: .
Similarly, we can use the formula used to solve the first part. and . Therefore, this sum is: .
Using these two answers, the desired sum is: .