show that the series 1 - 1/2³ + 1/3³ - 1/4³ + 1/5³ - .... is absolutely convergent
∑n=1∞(−1)n+1n31=1−231+331−431+531...
∣∑n=1∞(−1)n+1n31∣=∑n=1∞n31
we test if it is convergent using the integral test, in order to use the Integral Test the series terms MUST eventually be decreasing and positive, which is so with the given series
∫1∞f(x)dx
f(x)=x31
limt→∞∫1tx31dx=limt→∞−3x21∣1t
=limt→∞−3t21+3(1)21
=−∞1+31
=−0+31
=31
∵ the series is absolutely convergent
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