Question

Prove or disprove each of the following statements. Be sure to provide complete justification. \(a\) If \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) are both convergent series, then \(\sum_{n=1}^{\infty} a_n b_n\) is convergent. \(b\) If \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) are both divergent series, then \(\sum_{n=1}^{\infty} a_n b_n\) is divergent. \(c\) If \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) are both convergent series with positive terms, then \(\sum_{n=1}^{\infty} a_n b_n\) is convergent.

          Prove or disprove each of the following statements. Be sure to provide complete justification.
\(a\) If \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) are both convergent series, then \(\sum_{n=1}^{\infty} a_n b_n\) is convergent.
\(b\) If \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) are both divergent series, then \(\sum_{n=1}^{\infty} a_n b_n\) is divergent.
\(c\) If \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) are both convergent series with positive terms, then \(\sum_{n=1}^{\infty} a_n b_n\) is convergent.

Prove or disprove each of the following statements. Be sure to provide complete justification.
a If ∑n=1^∞ an and ∑n=1^∞ bn are both convergent series, then ∑n=1^∞ an bn is convergent.
b If ∑n=1^∞ an and ∑n=1^∞ bn are both divergent series, then ∑n=1^∞ an bn is divergent.
c If ∑n=1^∞ an and ∑n=1^∞ bn are both convergent series with positive terms, then ∑n=1^∞ an bn is convergent.

Added by Vicenta C.

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