【問題】有關SIGMA的問題

[X助攻萬歲X 10]

我在講義上看到一題:

3+33+333+3333+....至第n項的和為=?

我用3*(1+11+111+...)結果算不出來...

或者是3+(30+3)+(300+33)+....嗎?

<最討厭這種題目了...= ="">

[pianonrock 10]

剛剛沒看清楚,

let a0=1, a1=11,...an=10^n + an-1

s(0)= 3(a0) = 3

s(1)= 3(a0 + a1)= 3(1 + 11)

s(2) = 3(a0 + a1 + a2) = 3(1 + 11 + 111)

.

.

.

s(n) = 3(a0 + a1 + ...+an)= 3(1 + Σ (10^i + ai-1) ), i=1,2,3,...n

p.s. s(n) diverges,so there's no simple form for this, you can check this by various tests of divergence and convergence,

=).

[weiye 10]

3+33+333+3333+....

= 1/3 * {9+99+999+.....+n個九}

= 1/3 * {(10-1)+(100-1)+(1000-1)+.....(一後面有n個零 - 1) }

= 1/3 * {10^1 + 10^2 + .... 10^n - n個一}

= 1/3 *{前半塊用等比級數和公式 - n}

[weiye 10]

另解：

3+33+333+3333+....至第n項的和

= 3 + (3+30) + (3+30+300) + .... (3+30+300+...三後面有n個零)

=3*(n) + 30*(n-1)+300*(n-2) + .... + 三後面有n個零 * 1

因為是 Σ(等比)×(等差)，再用雜級數公式求和。

[weiye 10]

p.s. s(n) diverges,so there's no simple form for this, you can check this by various tests of divergence and convergence,

=).

題目沒有要求累加至無窮多項，只是要求至第n項的和(以 n 表示 S_n )。

如果要看收斂或發散，直接看第 n 項，因為 a_n → infinity，所以顯然 S_n 會發散。

[pianonrock 10]

discovergence of S(n) = discovergence of an, but by doing S(n), it would provide a more general understanding.=)

p.s. ratio tests provide lots of additional understanding to the subject, when you reach more advanced mathematics, the understanding of those in elementary calculus would be very important to get a better picture. so, it's a good idea to apply those techniques on every interesting problem you(we) encounter.=).

p.s.s. when "envolving" with mathematics, always try to get a more general picture instead of special one, that is, consider all the possibilities and the reasonings and motivations.=)

[weiye 10]

discovergence of S(n) = discovergence of an , but by doing S(n), it would provide a more general understanding.=)

it is not proper if you use "=" in the statement "discovergence of S(n) = discovergence of an".

since that the disconvergence of a_n does implies the disconvergence of S_n,

but it is not vice sersa. take a_n = 1/n, for instance, while a_n converges to 0 as n→infinity , but S(n)=1+1/2+..+1/n diverges.

what I say above is that you don't need to ues ratio test, root test, or comparision test (or even integral test, M-test .... etc.)

it's clear that the general term a_n = 333.....3333("3 appears n times") does diverge, so does S_n.

p.s. ratio tests provide lots of additional understanding to the subject, when you reach more advanced mathematics, the understanding of those in elementary calculus would be very important to get a better picture. so, it's a good idea to apply those techniques on every interesting problem you(we) encounter.=).

p.s.s. when "envolving" with mathematics, always try to get a more general picture instead of special one, that is, consider all the possibilities and the reasonings and motivations.=)

in the original question, X助攻萬歲X only ask to find out the sum of n terms, the partial sum, not the infinity sum, since it's clear that the infinity sum diverges.

[pianonrock 10]

sigh, I guess it's because of my poor intepretation =(, sorry i have been writting my paper till midnight everyday, thank you for fixing that =).

anyway, I thought the function of this board is not being a solution manual, but a place that provides additional materials and ideas that would help? =)

[X助攻萬歲X 10]

anyway, thanks a lot...(poor english)

也謝謝weiye~

[weiye 10]

放輕鬆，我也只是想提供給發問者知道

although "there's no simple form for" the infinity series, "there does exist simlple form for" the partial sum (finite series.)

擔心他看到你上面寫的

p.s. s(n) diverges,so there's no simple form for this, you can check this by various tests of divergence and convergence,

會誤解。

anyway, I thought the function of this board is not being a solution manual, but a place that provides additional materials and ideas that would help? =)

討論數學是有趣的，放輕鬆，呵呵，

討論的對象是數學，別想太多，呵呵。

sigh, I guess it's because of my poor intepretation =(, sorry i have been writting my paper till midnight everyday, thank you for fixing that =).

我以前半夜熬夜寫/想論文都會聽 Linkin Park 的歌，很有提神的作用，

提供給你可以試試，尤其是隔天還要 meeting the boss, 卻沒東西生出來的壓力的話，呵呵。

好像開始偏離數學了～～～（逃~~~:p）

[pianonrock 10]

lol so you doing your phd now? i am writting my paper, and try to apply math phd right after i graudate from univeristy, by the way, have you take Real Analysis? I AM SUFFERING NOW!!!= =mY school is using four texts at the same time, holy...

[weiye 10]

lol so you doing your phd now? i am writting my paper, and try to apply math phd right after i graudate from univeristy, by the way, have you take Real Analysis? I AM SUFFERING NOW!!!= =mY school is using four texts at the same time, holy...

以前大學跟研究所都修過 Real Analysis (不善長分析~還好都過了:p~)，

老師到沒用到四本課本，呵呵，不過教分析的老師用多本課本好像很常見，呵呵。

我猜猜哪四本，Royden, Zigmund, Rudin 有這三位作者寫的嗎？

[pianonrock 10]

忽最近很忙,..

不是耶 那些是我去年的..

我現在用的是ruidin的real and complex, conrway的functional analysis, folland的real analysis, 還有ruidin的functional analysis(supplement)