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Infinite Series convergence/divergence help?
sum of n=1 to infinity of 4/((2^n)+1). If it converges, find the sum, if it diverges, explai
1 Answer
- 9 years agoFavorite Answer
If we ignore the plus 1 in the denominator, we can do a little algebra...
4/(2^n)
= 4 * 1/(2^n)
= 4 * (1^n / 2^n)
= 4(1/2)^n
This is a geometric series which converges by geometric series test ( |1/2| < 1 ). Now, how to relate this to your problem. Notice, if you add 1 to the denominator, you decrease the value of the whole thing. What I mean is...
4/(2^n + 1) < 4/(2^n)
And since this is always true for all n, we can say that the your sum converges by direct comparison test.
