Question

87. Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

Accepted Answer

Understand the problem: We are asked to determine if the statement "If $$ \sum_{k=1}^{\infty} a_k $$ diverges, then $$ \sum_{k=10}^{\infty} a_k $$ also diverges" is true or false. Recall the definition of series convergence and divergence: A series $$ \sum_{k=m}^{\infty} a_k $$ converges if the sequence of partial sums $$ S_n = \sum_{k=m}^n a_k $$ approaches a finite limit as $$ n 	o \infty . $$Otherwise, it diverges. Analyze the relationship between the two series: The series starting at $$ k=10  is $$essentially the tail of the series starting at $$ k=1 . $$The original series can be written as $$ \sum_{k=1}^{\infty} a_k = \sum_{k=1}^{9} a_k + \sum_{k=10}^{\infty} a_k . $$Consider the impact of the finite sum $$ \sum_{k=1}^{9} a_k $$: Since this is a finite sum, it does not affect convergence or divergence of the infinite series. Therefore, the convergence or divergence of $$ \sum_{k=10}^{\infty} a_k $$ determines the behavior of the tail. Conclude based on the above: If the entire series $$ \sum_{k=1}^{\infty} a_k $$ diverges, then its tail $$ \sum_{k=10}^{\infty} a_k $$ must also diverge, because adding or removing a finite number of terms does not change the divergence property.

87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.

b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

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Key Concepts

Infinite Series and Convergence

Effect of Changing the Index of Summation

Counterexamples in Series Analysis