Calculus II · Sequences, series, and convergence tests
Expose exponential decay hidden in a changing power
Problem
Determine whether the positive-term series \[ \sum_{n=1}^{\infty}\left(\frac n{n+1}\right)^{n^2} \] converges. Apply the root test by computing the exact limit of the nth roots. State the standard exponential limit you use, and explain why observing only that the terms tend to zero would not settle the question.
Hint
Take the nth root before estimating the n-squared exponent.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
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