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© Copyright 1999, Jim Loy
Here is an infinite series which you may not have seen:
1/0!+1/1!+1/4!+1/5!+1/8!+1/9!+...
The sum very rapidly comes out to 2.05003... If you are handy with series, then you might want to try to figure out what the exact sum is (in terms of other functions). Those of you who have not dealt with series will probably never solve that. So, here is the solution. The sum is (e+sin1+cos1)/2. e is, of course, the base of natural logarithms, and has a value of 2.7182818284... The angle 1 is in radians.
How did I come up with this series? Here are three series which I think came from the Maclaurin series (I will deal with that in a future article):
ex=1+x+x2/2!+x3/3!+x4/4!+x5/5!+...
sin x=x-x3/3!+x5/5!-x7/7!+...
cos x=1-x2/2!+x4/4!-x6/6!+...
We make x=1 in all three equations, add the three equations together, and divide by 2, and we have my series. I was just messing around with various series, when I noticed that many of the terms cancelled each other out. By the way, (e-sin1-cos1)/2 is:
1/2!+1/3!+1/6!+1/7!+1/10!+1/11!+...
I received email saying that these series do come from Maclaurin series, but that, more basically, they come from power series. I will check on that eventually.
Addendum:
Here is another series, which I just made up (I am sure that it is previously known):
1/4+2/8+3/16+4/32+5/64...
Here is how I evaluate it using geometric series:
1/2 = 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
1/4 = 1/8 + 1/16 + 1/32 + 1/64 + ...
1/8 = 1/16 + 1/32 + 1/64 + ...
1/16 = 1/32 + 1/64 + ...
...
--------------------------------------------
1 = 1/4 + 2/8 + 3/16 + 4/32 + 5/64 + ...