Series Convergence Analysis

We need to determine the convergence of two series: $ \sum_{k=1}^{\infty} a_k $ and $ \sum_{k=1}^{\infty} b_k $, where $ a_k = 2^{-k}k^4 \sin k $ and $ b_k = 2^{-k^2} k \sin^2 k $. We will analyze each series separately using the Absolute Convergence Test and the Comparison Test.

Convergence of $ \sum_{k=1}^{\infty} a_k $

• Consider the absolute value of the terms: $ |a_k| = |2^{-k}k^4 \sin k| = 2^{-k}k^4 |\sin k| $.
• Since $ |\sin k| \leq 1 $, we have $ |a_k| \leq 2^{-k}k^4 $.
• We test the convergence of $ \sum_{k=1}^{\infty} 2^{-k}k^4 $ using the Ratio Test. Let $ c_k = 2^{-k}k^4 $.
• Calculate the limit: $ \lim_{k \to \infty} \left| \frac{c_{k+1}}{c_k} \right| = \lim_{k \to \infty} \frac{2^{-(k+1)}(k+1)^4}{2^{-k}k^4} = \lim_{k \to \infty} \frac{1}{2} \left( \frac{k+1}{k} \right)^4 = \frac{1}{2} (1)^4 = \frac{1}{2} $
• Since the limit $ \frac{1}{2} < 1 $, the series $ \sum_{k=1}^{\infty} 2^{-k}k^4 $ converges by the Ratio Test.
• By the Comparison Test, since $ |a_k| \leq 2^{-k}k^4 $ and $ \sum_{k=1}^{\infty} 2^{-k}k^4 $ converges, the series $ \sum_{k=1}^{\infty} |a_k| $ converges.
• Therefore, $ \sum_{k=1}^{\infty} a_k $ converges absolutely and hence converges.

Convergence of $ \sum_{k=1}^{\infty} b_k $

• Consider the absolute value of the terms: $ |b_k| = |2^{-k^2} k \sin^2 k| = 2^{-k^2} k \sin^2 k $.
• Since $ \sin^2 k \leq 1 $, we have $ |b_k| \leq 2^{-k^2} k $.
• We test the convergence of $ \sum_{k=1}^{\infty} 2^{-k^2} k $. Let $ d_k = 2^{-k^2} k $.
• Calculate the limit using the Ratio Test: $ \lim_{k \to \infty} \left| \frac{d_{k+1}}{d_k} \right| = \lim_{k \to \infty} \frac{2^{-(k+1)^2} (k+1)}{2^{-k^2} k} = \lim_{k \to \infty} \frac{2^{-k^2-2k-1} (k+1)}{2^{-k^2} k} $ $ = \lim_{k \to \infty} \frac{1}{2} \frac{k+1}{k} 2^{-2k-1} = \lim_{k \to \infty} \frac{1}{4} \left( 1 + \frac{1}{k} \right) 2^{-2k} $
• As $ k \to \infty $, $ 2^{-2k} \to 0 $. Thus, the limit is $ \frac{1}{4} \times 1 \times 0 = 0 $.
• Since the limit $ 0 < 1 $, the series $ \sum_{k=1}^{\infty} 2^{-k^2} k $ converges by the Ratio Test.
• By the Comparison Test, since $ |b_k| \leq 2^{-k^2} k $ and $ \sum_{k=1}^{\infty} 2^{-k^2} k $ converges, the series $ \sum_{k=1}^{\infty} |b_k| $ converges.
• Therefore, $ \sum_{k=1}^{\infty} b_k $ converges absolutely and hence converges.

Conclusion

Both series $ \sum_{k=1}^{\infty} a_k $ and $ \sum_{k=1}^{\infty} b_k $ converge.