Use the ratio test to determine whether the series is convergent or divergent.
1/2 + 2^2/2^2 + 3^2/2^3 + 4^2/2^4 + ...
As n-> infinity, the ratio of successive terms, [(n+1)/n]^2*(1/2), approaches 1/2
Read about the ratio test here (if you are not already familiar with it):
http://en.wikipedia.org/wiki/Ratio_test
That will tell you what the answer is in this case.
To determine whether the series is convergent or divergent, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series is convergent. If the limit is greater than 1, the series is divergent. Finally, if the limit is equal to 1, the test is inconclusive, and we need to use another test.
Let's apply the ratio test to the given series. The general term of the series is given by:
a_n = n^2 / 2^n
To apply the ratio test, we need to calculate the limit as n approaches infinity of the absolute value of (a_(n+1) / a_n). Let's do that:
lim (n->∞) |(a_(n+1) / a_n)|
= lim (n->∞) |((n+1)^2 / 2^(n+1)) / (n^2 / 2^n)|
= lim (n->∞) |((n+1)^2 / 2^(n+1)) * (2^n / n^2)|
= lim (n->∞) |((n+1)^2 * 2^n) / (2^(n+1) * n^2)|
= lim (n->∞) |((n+1)^2 * 2^n) / (2^n * 2 * n^2)| [Dividing the exponents]
Now, simplify the expression further:
= lim (n->∞) |(n^2 + 2n + 1) / (2n^2)| [Canceling out the 2^n terms]
Next, let's take the limit of this expression:
= lim (n->∞) |(1 + 2/n + 1/n^2) / 2| [Dividing each term by n^2]
Now, as n approaches infinity, both 2/n and 1/n^2 will approach zero, so we can simplify the expression even more:
= lim (n->∞) |(1 + 0 + 0) / 2|
= |1/2|
Since the absolute value of the limit is less than 1, we can conclude that the series is convergent according to the ratio test.