- #1

mynameisfunk

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**limit proofs(indeterminate forms??)**

## Homework Statement

We work in the real numbers. Are the following true or false? Give a proof or counterexample.

(a) If [itex]\sum a^4_n[/itex] converges, then [itex]\sum a^5_n[/itex] converges.

(b) If [itex]\sum a^5_n[/itex] converges, then [itex]\sum a^6_n[/itex] converges.

(c) If [itex]a_n \geq 0[/itex] for all [itex]n[/itex], and [itex]\sum a_n[/itex] converges, then [itex]na_n \rightarrow 0[/itex] as [itex]n \rightarrow \infty[/itex].

(d) If [itex]a_n \geq 0[/itex], for all [itex]n[/itex], and [itex]\sum a_n[/itex] converges, then [itex]n(a_n - a_{n-1}) \rightarrow 0[/itex] as [itex]n \rightarrow \infty[/itex].

(e) If [itex]a_n[/itex] is a decreasing sequence of positive numbers, and [itex]\sum a_n[/itex] converges, then [itex]na_n \rightarrow 0[/itex] as [itex]n \rightarrow \infty[/itex].

## Homework Equations

## The Attempt at a Solution

(a) and (b) can be proved similarly. Since [itex]\sum a_n^4[/itex] converges, for some [itex]N[/itex], when [itex]n \geq N[/itex], then [itex]a_n^4 < 1[/itex]. Take [itex]\beta[/itex] s.t., [itex]a_n^4 < \beta < 1[/itex]. That is, [itex]a_n < (\beta)^{1/4} < 1[/itex]. Also, [itex]|(\beta)^{1/5}| < 1[/itex]. This implies that [itex]|a_n^5| < 1[/itex] and therefore [itex]\sum |a_n^5|[/itex] converges.

(c) This is where I get confused. THis seems like an indeterminate form, we have never done this in class. same for (d) and (e). any suggestions?