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What is Calculus?

Calculus is the study of what happens if you let āˆž\infty into the picture.

What do you get if you add up an infinite number of fractions?

12+14+18+116+132+164+1128+1256+1512+11024+12048+...\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \frac{1}{128} + \frac{1}{256} + \frac{1}{512} + \frac{1}{1024} + \frac{1}{2048} + ...

Not just a lot of terms, but infinite terms? You might think that if you add up infinitely many positive terms you get infinity, but that's not true! The sum above is actually equal to exactly 1.

That might be surprising, but maybe this picture will convince you:

Geometric proof that the infinite sum is equal to 1

Calculus is the study of situations like this, where we allow for the concept of infinity and get exact answers as a result.