Assuming the pattern that you observed continues forever—and it does—the sum $O$ of the first $n$ odd numbers is given by the formula: $O=1+3+…+2n-1=n^2$. This fact can be proved using Mathematical Induction. Adding $n$ is the same thing as adding $1$ $n$ times. So we obtain: $E=2+4+…+2n=n^2 +n$. From this it is easy to find the sum of the first $n$ numbers. We divide each term by $2$ to get $$1+2+3+…+n=(n^2 +n)/2 = n(n+1)/2.$$
Once again Sarthak asked some deep questions on which today’s experiments are based. He asked: What meaning can be given to division by 0?
At the outset, let me say that division by 0 is not allowed.
The reason is the confusion caused by the following argument: We know that $0=0.$ This implies that $0$ times $1$ is equal to $0 $ times $2$. So $0 \times 1 = 0 \times 2$. Now canceling $0$ from both sides (by dividing both sides with $0$) we obtain $1=2$, which clearly is false. And there is a theorem in mathematics that says that a false proposition implies any proposition. Which implies that Santa Claus exists. And also implies that Santa Claus does not exist.
Now you can see why mathematicians have very sensibly banned division by $0$.
However, this does not mean that mathematicians do not try to give division by $0$ some meaning. Try the following (you may need a calculator or a computer for some of these experiments):
However, this does not mean that mathematicians do not try to give division by $0$ some meaning. Try the following (you may need a calculator or a computer for some of these experiments):
1. Make a table of the function $y=1/x$. In the first column place numbers very close to $0$ that are positive. For example $0.1, 0.01, 0.001, \dots, 0.0000000001$. In the second column, replace $x $ in $1/x$ by this number and see what values come out for y. For example, when we replace $x$ by $.01$, we get $y=100. $ Describe the results of your experiments.
2. Now replace $x$ by negative numbers that come closer and closer to $0$ and find out what value $ y$ takes. Describe the results of your experiments.
3. Try the above with the function $y=x^2/x$. Does the value come closer and closer to a number?
2. Now replace $x$ by negative numbers that come closer and closer to $0$ and find out what value $ y$ takes. Describe the results of your experiments.
3. Try the above with the function $y=x^2/x$. Does the value come closer and closer to a number?
Describe the results of your experiments.
We say a function (depending on the variable $x$) approaches the limit $+\inf$ (read plus infinity) when $x$ approaches a number $a$ from the left, if given any large number $M$ (any large number you can think of), we can find a number a little smaller than $a$, so that the corresponding value of the function becomes larger than $M$. (The ‘left’ refers to the number being to the left of $a$ on the number line.) Similarly, we can define the limit approaching from the right, and the limit approaching minus infinity.
We say a function (depending on the variable $x$) approaches the limit $+\inf$ (read plus infinity) when $x$ approaches a number $a$ from the left, if given any large number $M$ (any large number you can think of), we can find a number a little smaller than $a$, so that the corresponding value of the function becomes larger than $M$. (The ‘left’ refers to the number being to the left of $a$ on the number line.) Similarly, we can define the limit approaching from the right, and the limit approaching minus infinity.
Intuitively, it is easier to understand the limit of a function if it approaches a number $q$, when $x$ approaches a number $a$. We replace $x$ by numbers close to $a$, and find the value of the function. If the value comes close to a number $q$, the limit is $q$. Try the following experiments and get a feel for the definition.
4. Multiply $(1-x)$ in turn by $(1+x),$ $(1+x+x^2),$ $(1+x^2+x^3)$ and simplify. Can you generalize the pattern?
5. What is the limit of $(1-x^2)/(1-x),$ $(1-x^3)/(1-x),$ $\dots$, $(1-x^n)/(1-x),$ as $x$ approaches $1$. (Hint: Replace $x$ by numbers close to $1$, like $1, 1.1, 0.9, 1.01, 0.99, 1.001, 0.999,$ $\dots,$ and guess the answer in each case.)
4. Multiply $(1-x)$ in turn by $(1+x),$ $(1+x+x^2),$ $(1+x^2+x^3)$ and simplify. Can you generalize the pattern?
5. What is the limit of $(1-x^2)/(1-x),$ $(1-x^3)/(1-x),$ $\dots$, $(1-x^n)/(1-x),$ as $x$ approaches $1$. (Hint: Replace $x$ by numbers close to $1$, like $1, 1.1, 0.9, 1.01, 0.99, 1.001, 0.999,$ $\dots,$ and guess the answer in each case.)
