Suppose summation an and summation bn are series with positive terms and summation bn is known to be divergent. If an&lt;bn for all n, what can you say about summation an? Why?

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Suppose summation an and summation bn are series with positive terms and summation bn is known to be divergent. If an&amp;lt;bn for all n, what can you say about summation an? Why?

Debbie Moore · Accepted Answer

Step 1  There is nothing that we can say about the series ∑an It can both converge and diverge as illustrated by the next example:  Let bn=1n. We know that ∑bn is a divergent series. First, let an=1n2. Now, we have an&amp;lt;bn for all n≥2, and we know that ∑an converges.  Next, let an=1n+1 Again, we know that an&amp;lt;bn for all n. Using the Limit Comparison Test, we can show that ∑an diverges:  limn⇒∞anbn=limn⇒∞1n+11n=limn⇒∞nn+1=limn⇒∞nnnn+1n  =limn⇒∞11+1n=limn⇒∞1limn⇒∞1+limn⇒∞1n=11−0=1  From here, we can conclude that ∑an is a divergent sum  Therefore, we have shown that ∑an can be both a convergent and a divergent series - thus, we cannot say anything about it in this case.

Buck Henry · Accepted Answer

Step 1  We cannot conclude convergence/devirgence of ∑an from the given information  Step 2  Example where ∑an converges  Let an=1n2 and bn=1n  Both an and bn are positive for n≥1  1n2&amp;lt;1n for n≥1  ∑n=1∞bn=∑n=1∞1n deverges  ∑n=1∞an=∑n=1∞1n2 converges  Step 3  Example where an diverges  Let an=12n and bn=1n  Both an and bn are positive for n≥1  12n&amp;lt;1n for n≥1  ∑n=1∞bn=∑n=1∞1n diverges  ∑n=1∞an=∑n=1∞12n diverges

Suppose summation a_{n} and summation b_{n} are series with positive

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