Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

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Ricky
2006-01-29 03:40:13

ii. By the ratio test:

Since
<
for all n:

< 1

Which means that the series converges, although I know of no tests that actually find this number.

And why isn't the latex '<' not working?

God
2006-01-29 02:22:04

Thanks. I had a typo on part iii though:
I meant such that ƒ(x+1) = sin( ƒ(x) )

ryos
2006-01-28 18:15:25

i. I think so.
ii. It stands to reason that, although the numbers will get so small as to be essentially 0, there will always be some teeny tiny insignificant bit more to add. I wouldn't know how to find out, though.
iii. Sure there is: ƒ(x) = sin(x-1)

God
2006-01-27 09:12:46

Say we define a sequence such that:
a_0 = 1
and for each successive term;
a_n+1 = sin(a_n)

i. The infinite limit of the sequence a_n is 0, right?
ii. Does this infinite series have a finite sum? If so, what is it?
iii. Is there any explicit function ƒ(x) such that ƒ(x+1) = sin( ƒ(x) )?

-thanks