11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰

Verified step by step guidance

1. Identify the general term of the series: \(a_k = \frac{10^k + 1}{k^{10}}\).
2. Observe the behavior of the numerator and denominator separately: the numerator grows exponentially as \$10^k\(, while the denominator grows polynomially as \)k^{10}$.
3. Recall that exponential growth dominates polynomial growth, so \$10^k\( grows much faster than \)k^{10}$ as \(k \to \infty\).
4. Apply the Divergence Test (also known as the Test for Divergence) by checking the limit of \(a_k\) as \(k\) approaches infinity: calculate \(\lim_{k \to \infty} \frac{10^k + 1}{k^{10}}\).
5. Since the numerator grows exponentially and the denominator polynomially, this limit does not approach zero; therefore, by the Divergence Test, the series diverges.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity.

Comparison Test

The Comparison Test involves comparing a given series to a second series with known convergence behavior. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.

Behavior of Exponential vs. Polynomial Terms

Exponential terms like 10^k grow much faster than polynomial terms like k^10. When analyzing series terms involving both, the exponential growth dominates, often causing divergence despite the polynomial denominator.