Question
Prove that if $\Sigma a_{n}$ diverges and $\Sigma b_{n}$ converges, then $\Sigma\left(a_{n}+b_{n}\right)$ diverges.
Step 1
Step 1: Assume for contradiction that $\Sigma\left(a_{n}+b_{n}\right)$ converges. Show more…
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Key Concepts
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Given two infinite series $\Sigma a_{n}$ and $\Sigma b_{n}$ such that $\Sigma a_{n}$ converges and $\Sigma b_{n}$ diverges, prove that $\Sigma\left(a_{n}+b_{n}\right)$ diverges.
Infinite Series
Series and Convergence
Proof Given two infinite series $\Sigma a_{n}$ and $\Sigma b_{n}$ such that $\sum a_{n}$ converges and $\Sigma b_{n}$ diverges, prove that $\Sigma\left(a_{n}+b_{n}\right)$ diverges.
Proof Suppose that $\Sigma a_{n}$ and $\Sigma b_{n}$ are series with positive terms. Prove that if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\infty$ and $\Sigma b_{n}$ diverges, then $\Sigma a_{n}$ also diverges.
Comparisons of Series
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