converges then 
must also be convergent.Mini quiz #4
(Oct.27.2000)
TRUE OR FALSE ? (do not justify your answer)
1. If
converges then
must also be convergent.
TRUE.
convergent means that 
converges for x=6. The centre of the interval of convergence of
is at 0. Since x=6 is in the interval of convergence, the radious of convergence
is at least 6. So, -2 must be in the interval of convergence, so that
must converge too.
2. If
converges then 
must also be convergent.
FALSE. A counter example is the series
which converges, yet 
does not converge.
3. If
diverges when x=6 then it diverges when x=10.
TRUE. If
converges for x=10 then it would converge for x=6 too (sam argument as in
1. above)
4. 
could be the interval of convergence of a power series.
TRUE. One example is the series 
5. 
is never equal to 
.
FALSE. Take all 
to be 0, say.