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I was thinking the following: A sum of an infinite number of values, no matter how big, but positive equals infinite.

And then I discovered the following row:

$\sum_{n=0}^\infty (\frac{1}{2^{n}})$

This is: $ 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...$

When I started calculating it with a Java application, I think I will never reach $2$. Check out the output of my application: (use the scrollbar, I aligned the numbers)

1/00000000000000000000000000000000000000000000000000000000000000000000000000000001: 1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000002: 1.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000004: 1.7500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000008: 1.8750000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000016: 1.9375000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000032: 1.9687500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000064: 1.9843750000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000128: 1.9921875000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000256: 1.9960937500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000000512: 1.9980468750000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000001024: 1.9990234375000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000002048: 1.9995117187500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000004096: 1.9997558593750000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000008192: 1.9998779296875000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000016384: 1.9999389648437500000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000032768: 1.9999694824218750000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000065536: 1.9999847412109375000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000131072: 1.9999923706054687500000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000262144: 1.9999961853027343750000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000000524288: 1.9999980926513671875000000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000001048576: 1.9999990463256835937500000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000002097152: 1.9999995231628417968750000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000004194304: 1.9999997615814208984375000000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000008388608: 1.9999998807907104492187500000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000016777216: 1.9999999403953552246093750000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000033554432: 1.9999999701976776123046875000000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000067108864: 1.9999999850988388061523437500000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000134217728: 1.9999999925494194030761718750000000000000000000000000000000000000000000000000000000000000000000000000
1/00000000000000000000000000000000000000000000000000000000000000000000000268435456: 1.9999999962747097015380859375000000000000000000000000000000000000000000000000000000000000000000000000

... a bit further in the output:

1/00003618502788666131106986593281521497120414687020801267626233049500247285301248: 1.9999999999999999999999999999999999999999999999999999999999999999999999999997236426062369777719876367
1/00007237005577332262213973186563042994240829374041602535252466099000494570602496: 1.9999999999999999999999999999999999999999999999999999999999999999999999999998618213031184888859938183
1/00014474011154664524427946373126085988481658748083205070504932198000989141204992: 1.9999999999999999999999999999999999999999999999999999999999999999999999999999309106515592444429969091
1/00028948022309329048855892746252171976963317496166410141009864396001978282409984: 1.9999999999999999999999999999999999999999999999999999999999999999999999999999654553257796222214984545
1/00057896044618658097711785492504343953926634992332820282019728792003956564819968: 1.9999999999999999999999999999999999999999999999999999999999999999999999999999827276628898111107492272
1/00115792089237316195423570985008687907853269984665640564039457584007913129639936: 1.9999999999999999999999999999999999999999999999999999999999999999999999999999913638314449055553746136
1/00231584178474632390847141970017375815706539969331281128078915168015826259279872: 1.9999999999999999999999999999999999999999999999999999999999999999999999999999956819157224527776873068
1/00463168356949264781694283940034751631413079938662562256157830336031652518559744: 1.9999999999999999999999999999999999999999999999999999999999999999999999999999978409578612263888436534
1/00926336713898529563388567880069503262826159877325124512315660672063305037119488: 1.9999999999999999999999999999999999999999999999999999999999999999999999999999989204789306131944218267

Before the colon, I print the last fraction that is added to the sum and after the colon, I have the current sum.

So, does this mean my first statement isn't correct?

Arturo Magidin
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Martijn Courteaux
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    Your first statement is incorrect (but don't feel too bad: Zeno argued the same thing, and noted it led to a paradox). The partial sums are always strictly smaller than $1$, but the limit of the partial sums is equal to exactly $1$. An infinite sum of positive quantities may grow without bound ("be infinite"), but it may also have a finite limit. See http://math.stackexchange.com/questions/29023/value-of-sum-xn/29035#29035 – Arturo Magidin May 06 '11 at 20:21
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    On the other hand, the sum of an infinite uncountable collection of positive numbers is always infinite. – Did May 06 '11 at 20:49
  • Concerning Zeno's paradoxes they are discussed in this Wikipedia entry http://en.wikipedia.org/wiki/Zeno's_paradoxes – Américo Tavares May 06 '11 at 21:27

2 Answers2

11

Yes, your first statement was incorrect. As long as the positive values $a_n$ you are summing decrease to 0 fast enough, the sum $\sum_{n=0}^\infty a_n$ will be finite even though there are infinitely many of the $a_n$'s. However, the question of what "fast enough" exactly means is a subtle one; for example $$\sum_{n=1}^\infty \frac{1}{n}\text{ diverges (i.e., “ = $\infty$”)}$$ so $\frac{1}{2},\frac{1}{3},\ldots$... does not go to 0 fast enough. Series can be determined to converge (i.e., equal a finite value) or diverge (i.e., become arbitrarily large, which metaphorically is like the sum being infinite) using convergence tests.

The thing to remember about infinite series is that they are just a particular kind of limit. The definition of the infinite sum $$\sum_{n=1}^\infty a_n$$ is $$\lim_{N\rightarrow\infty}\,\,\,\,\sum_{n=1}^N a_n$$ Algebraically, i.e. solely with the operation "$+$", there is no such thing as an infinite sum. The infinite sum can only be defined using a limit. Why is this important?

Limits can be different from the numbers you are taking the limit of. For example, the limit of the sequence $\frac{1}{2},\frac{3}{4},\frac{7}{8},\ldots,\frac{2^n-1}{2^n},\ldots$ is 1, even though none of the entries of the sequence are equal to 1.

So, the fact that none of the partial sums of $\sum_{n=0}^\infty\frac{1}{2^n}$, i.e. the terms $\sum_{n=0}^N \frac{1}{2^n}$ that we are really taking the limit of, are equal to 2, does not prevent the infinite sum (i.e., the limit of the partial sums) from being 2.

Also, thinking about infinite sums as limits explains why we have problems with them that we didn't have when talking about finite sums. For example, the infinite sum $$\sum_{n=0}^\infty (-1)^n = 1 -1 +1 -1 + \cdots $$ does not exist, because (using the proper definition) $$\sum_{n=0}^\infty (-1)^n = \lim_{N\rightarrow\infty} \quad\sum_{n=0}^N (-1)^n = \lim_{N\rightarrow\infty} (\text{1 if $N$ even, 0 if $N$ odd})$$ and the limit of this oscillating sequence does not exist.

Zev Chonoles
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1

Your statement is not correct. I was about to write an elaborate answer when I figured out that you should read http://en.wikipedia.org/wiki/Geometric_series in any case.

The sum $\displaystyle\sum_{i=0}^\infty a^i$ where $|a| < 1$ converges to $\frac{1}{1-a}$. This follows from the identity $$(1+a+ a^2 + \ldots + a^n)(1-a) = 1 - a^{n+1}.$$

M.B.
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