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Calculus in a Nutshell

The concept of infinity often causes problems and paradoxes when it appears in math and science applications.

Calculus helps tame infinity; essentially, this is the purpose of calculus.

In this quiz, we'll encounter infinity (∞) (\infty) for the first time by exploring some features of infinite sums, which we mentioned in the last quiz.

Infinite Sums in a Nutshell

                   
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We need some new language to express what we mean by an infinite sum. To start us off, let's learn how to write out sums with many terms.

For example, writing out every term in the sum of 1,12,13,…, 1 , \frac{1}{2}, \frac{1}{3} , \dots , all the way up to 1100\frac{1}{100} would take a lot of space and effort! Instead, we write

1+12+13+⋯+1100=∑j=11001j. 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{100} = \sum\limits_{j=1}^{100} \frac{1}{j} .

The letter Sigma ∑ \sum is commonly used to represent sums in math and science. It's helpful whenever we sum a long list of numbers, like above.

Generally, if aj a_{j} is the jthj^{\text{th}} number in our list, then

∑j=1naj=a1+a2+⋯+an; \sum_{j=1}^{n} a_{j} = a_{1} + a_{2} + \dots + a_{n};

the "j=1j=1" at the bottom of ∑ \sum tells us the smallest value of the integer j,j, and the "nn" on top is jj's largest value; all the aja_{j}'s are summed up.

To get some practice with the notation, evaluate ∑j=14j2. \sum\limits_{j=1}^{4} j^2.

Infinite Sums in a Nutshell

                   
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To study an infinite sum, we first look at a finite sum with a variable as a stopping index, as in the sum of the first nn consecutive positive integers ∑j=1nj. \sum\limits_{j=1}^{n} j. The sum depends on nn: for example, if n=4, n = 4, we find 1+2+3+4=10, 1+2 + 3 + 4 =10, but if n=5, n = 5, we get 1+2+3+4+5=15. 1 +2 + 3 + 4 + 5 = 15.

Sometimes we get lucky and there's a formula that lets us sidestep adding all the terms in a sum once we give nn a value. For example, ∑j=1nj=n(n+1)2. \sum\limits_{j=1}^{n} j = \frac{n(n+1)}{2}. Use this to find the sum of the first 200 consecutive positive integers 1+2+3+4+⋯+198+199+200. 1 + 2 + 3 + 4 + \dots + 198+199 + 200.

Infinite Sums in a Nutshell

                   
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While we can't add up an infinite list a1,a2,…, a_{1}, a_{2} , \dots , we can do the sums ∑j=1naj \sum_{j=1}^{n} a_{j} for any finite nn if we have enough resources (and patience!).

We define ∑j=1∞aj=L \sum_{j=1}^{\infty} a_{j} = L to mean that the partial sums ∑j=1naj \sum_{j=1}^{n} a_{j} tend to get closer and closer to L L as n n gets larger and larger.

Here, LL stands for some real number. Let's look at two examples: ∑j=1∞910jand∑j=1∞9×10j. \sum\limits_{j=1}^{\infty} \frac{9}{10^{j}} \hspace{0.5cm} \text{and} \hspace{0.5cm} \sum\limits_{j=1}^{\infty} 9 \times 10^{j}. One of these has meaning as a real number, the other does not.

Which one of these infinite sums actually exists as a real number?

Hint: Note that 0.9ˉ 0.\bar{9} (a decimal point followed by an infinite number of 9's) is the same thing as 1. Check out this wiki on why this is so.

Infinite Sums in a Nutshell

                   
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We say that the infinite sum converges when we can find L L so that ∑j=1∞aj=L \sum\limits_{j=1}^{\infty} a_{j} = L in the sense given in the last problem. When this is impossible, we say the infinite sum diverges.

Determining whether a sum converges or diverges is a subtle business, and a large chunk of calculus is devoted to finding convergence tests.

This course won't cover these tests, but we will glimpse a few interesting applications of infinite sums. While many of these involve converging sums, diverging sums play important roles, too, as we're about to see.

Infinite Sums in a Nutshell

                   
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To see some of the quirks of diverging sums and their uses in the real world, let's consider the Tower of Lire puzzle.

We begin with a stack of two blocks. The goal here is to keep the tower from toppling while making the topmost reach beyond the vertical line.

If the center of mass of the top block goes beyond the edge of the bottom one, the tower falls; if the center of mass of the two blocks together extends over the edge of the pallet, the tower falls.

The top circle in the interactive below tracks the center of mass of the top block, while the bottom circle tracks the center of mass of the pair. The former will change from blue to red if it goes beyond the edge of the lower block; the latter changes if it goes beyond the edge of the palette.

Is it possible to position the blocks so that the topmost extends beyond the vertical line while keeping the tower from falling?

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Suppose we move the line a bit to the right. Is it still possible to have the topmost block overreach the vertical line without the tower falling?

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Let's add a third block to the tower from the last problem.

Is it possible to arrange the blocks so that the topmost overreaches the vertical line without falling?

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To summarize the last few problems, if we make the Tower of Lire tall enough, we can arrange the blocks to overhang any horizontal distance.

We'll prove that the maximum overhang of nn blocks is ∑j=1n1j,\sum_{j=1}^{n}\frac{1}{j}, which is approximately ln⁡(n) \ln(n) when nn is large enough.

Using either the intuition of the Tower of Lire puzzle or this formula for the maximum overhang, make a statement about the harmonic sum ∑j=1∞1j. \sum\limits_{j=1}^{\infty} \frac{1}{j}.

Infinite Sums in a Nutshell

                   
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Since adding an infinite list of numbers is generally impossible, ∑j=1∞aj=L \sum\limits_{j=1}^{\infty} a_{j} = L means that the partial sums ∑j=1naj \sum_{j=1}^{n} a_{j} get closer and closer to LL as nn gets larger and larger, or closer and closer to ∞ \infty if you like.

In general, if some quantity f f depends on some variable xx (like the partial sums depend on nn), we write lim⁡x→af(x)=L \lim\limits_{ x \to a} f(x) = L to mean ff gets closer and closer to L L as x x gets closer and closer to a.a. Here, "lim⁡ \lim" is short for limit, which is our primary tool for taming ∞. \infty.

For example, we can rewrite the infinite sum expression above as lim⁡n→∞∑j=1naj=L. \lim\limits_{n \to \infty} \sum\limits_{j=1}^{n} a_{j} = L.

Defining derivatives, integrals, and infinite sums is impossible without confronting ∞, \infty, so limits sit at the heart of all calculus.

In the next chapter, we'll learn all of the essential ideas behind this crucial concept.

Infinite Sums in a Nutshell

                   
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