Name: Anonymous 2006-06-23 14:35
1 = 1
1+4 = 5
1+4+16 = 21
1+4+16+64 = 85
...and so forth. Each time we add a term, it's the previous sum plus an even larger number. With 13 terms, it's already up to about 22 million. And yet, if we add enough of these series together, we should get -1/3.
Why?
Taylor series expansion for 1/(1-x^2) is
SUM (2^(2n)) for n=0 to n = infinity. The earlier we stop (lower n) the sum. So plug in 2. The numbers in the series sum are 1, 4, 16, 64, etc. It's always going to get bigger. But what about plugging in the value directly.
2^2 = 4
1-4 = -3
1/ (-3) = -1/3
Hence, infinity = -1/3? But if you try it with another number, you get another series. What's going on here?
1+4 = 5
1+4+16 = 21
1+4+16+64 = 85
...and so forth. Each time we add a term, it's the previous sum plus an even larger number. With 13 terms, it's already up to about 22 million. And yet, if we add enough of these series together, we should get -1/3.
Why?
Taylor series expansion for 1/(1-x^2) is
SUM (2^(2n)) for n=0 to n = infinity. The earlier we stop (lower n) the sum. So plug in 2. The numbers in the series sum are 1, 4, 16, 64, etc. It's always going to get bigger. But what about plugging in the value directly.
2^2 = 4
1-4 = -3
1/ (-3) = -1/3
Hence, infinity = -1/3? But if you try it with another number, you get another series. What's going on here?