级数求和下面两个都对吗？

which of the following are correct? 1) both a and b, 2) only a, 3) only b, 4) neither a nor b
a) Solve S.
S = 1-1+1-1+1..... S = 1 -(1-1+1-1+1.....) S = 1 - S S = 1/2
b)  Given pi^2 / 8 = 1/1^2 + 1/3^2 + 1/5^2 + 1/7^2+... Solve S.
S = 1/1^2 + 1/2^2+ 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2+... S = (1/1^2 + 1/3^2 + 1/5^2 + 1/7^2+...) +   (1/2^2+ 1/4^2 + 1/6^2 + 1/8^2+...)
S = pi^2 / 8 + 1/4 * S S = pi^2 / 6

回复 1楼 microsat 的帖子
a) is incorrect, b) is correct

3770Keystone 发表于 2025-06-27 14:29
回复 1楼 microsat 的帖子
a) is incorrect, b) is correct

谢谢！
同样是代入法，为何a是错误的呢？
能详细说说吗？

microsat 发表于 2025-06-27 13:31
which of the following are correct? 1) both a and b, 2) only a, 3) only b, 4) neither a nor b
a) Solve S.
S = 1-1+1-1+1..... S = 1 -(1-1+1-1+1.....) S = 1 - S S = 1/2
b)  Given pi^2 / 8 = 1/1^2 + 1/3^2 + 1/5^2 + 1/7^2+... Solve S.
S = 1/1^2 + 1/2^2+ 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2+... S = (1/1^2 + 1/3^2 + 1/5^2 + 1/7^2+...) +   (1/2^2+ 1/4^2 + 1/6^2 + 1/8^2+...)
S = pi^2 / 8 + 1/4 * S S = pi^2 / 6

标题看用户名系列
绝对收敛了解一下

回复 3楼 microsat 的帖子
收敛和不收敛的区别

lzhiyang 发表于 2025-06-27 15:21
回复 3楼 microsat 的帖子
收敛和不收敛的区别

只能收敛的才能用代入法？
那一般怎么快速的证明一个级数和是收敛的呢？

回复 6楼 microsat 的帖子
微積分2

这个人是faculty？难怪美帝药丸

hrshidsb 发表于 2025-06-27 15:57
这个人是faculty？难怪美帝药丸

LZ 还审过稿呢

楼主还是统计类的faculty 难以想象能问出这问题？ 不过楼主性格挺好的，大家说他也不会生气。

answer 3)

microsat 发表于 2025-06-27 15:05
谢谢！
同样是代入法，为何a是错误的呢？
能详细说说吗？

Many rules for finite sums may fail for infinite sums. Using them blindly could lead to mistakes such as (a) which is itself a divergent series (meaning it has no sum). It is also well-known that a conditionally convergent series can be rearranged so that it would converge to any value you want (or become divergent), e.g. 1/1-1/2+1/3-1/4+1/5 -.... The series in (b), on the other hand, is absolutely convergent and behaves much more predictably.