98% Students Cannot Solve This

Today we will look at this beautiful infinite series. We need to find the value of the sum from n equals zero to infinity of n divided by n plus one factorial plus n factorial. At first glance, this looks complicated, but we will slowly break it down and see a hidden pattern. As a first step, let us simplify the denominator. Notice that n plus one factorial can be written as n plus one times n factorial. Now we can take the n factorial common from both terms. So the denominator becomes n factorial times n plus one plus one, which is n factorial times n plus two. Good. Now we will make a smart move. We will rewrite n in the numerator as n plus two minus two. This is done on purpose to break the fraction into two simpler parts. So now we will split this into two parts. The first part becomes n plus two divided by n factorial times n plus two. The second part becomes minus two divided by n factorial times n plus two. Now look carefully at the first part. n plus two in numerator and denominator cancel, leaving one divided by n factorial. Now focus on the second sum. Multiply and divide by n plus one. So this can be rewritten as two times n plus one divided by n plus two factorial. This numerator can be written as two n plus two. Now again, we will do some manipulation. Rewrite this two as four minus two. Then split the fraction like this. Now in this part, take two as common to get this as two times n plus two. So this becomes two divided by n plus one factorial. Now expand the bracket to get minus of this sum plus this sum. Take two outside from both the sums. Does this look familiar? Yes. Recall the Taylor series expansion of e raised to x, which is this. If we put x as one, we get this. Now look at this part. This is nothing but e. Then look at this sum. It can be written as this plus this and so on. So it can be written as e minus one. Then look at this sum. It can be written as this plus this and so on. So it can be written as e minus one minus one over one factorial. So it becomes e minus two. Expand this to get this. Then expand this to get this. Thus we get e minus two. So finally, the value of the given infinite series is e minus two. And that's it. Like, share and subscribe.