The fundamental theorem of Calculus:
The fundamental theorem of calculus asserts the interrelated properties of integration and differentiation. It says that a function when differentiated, can be brought back by integrating (anti-derivative) or a function when integrated, can be brought back by differentiation.
First theorem: Let f be a function that is integrable on [a,x] for each x in [a,b], then let c be such that a≤c≤b and define a new function A as follows,
A(x)=∫_c^x▒f(x)dt, if a≤x≤b.
Then the derivative A’(x) exists at each point x in the open interval (a,b) where f is continuous, and for such x, A^' (x)=f(x).
This can be intuitively understood, by thinking of integration and differentiation as operations that work opposite to each …show more content…
According to the definition, if we progress to a certain term in sequence f(n), where it is sufficiently close to our limit L, then there difference of the two is less compared to a very small band of thickness ɛ. And when we surpass this certain value of N (which we define as an arbitrary threshold) to a greater value n, our difference becomes smaller and smaller until it is very close to or equal to zero. Therefore, very gallantly, we can say that at some point, our sequence will converge. And if it does not, then we can proceed to say that it diverges- that is the exact opposite, where the difference keeps getting bigger and …show more content…
We saw that the general term of our sequence was 1/n. So, as we keep increasing the value of n and eventually to infinity, the value of 1/n becomes zero. Thus, we fix that as our limit.
|f(n)-L|<ɛ, [L=0],
|f(n)|<ɛ,
Therefore, 1/n<ɛ, implies that, 1/ɛ<n.
We know that n≥N.
So, 1/ɛ<N.
Finally, we can claim that the sum of an infinite series is not just a distinct algebraic sum of the terms, but rather the “sum”, is a very special sum. It is obtained as the limit of the sequence of partial sums. The very concept of summation becomes hesitant when we progress further where the terms themselves become hazy for a clear operation. A ‘sum’ itself is a number and is just as it is, according to face value. The difference is how the sum manifests itself in a ‘series’, where we see whether it is converging or diverging and notice it moving across the plane, stopping and meandering.
Reference: T.M. Apostol: One variable Calculus. The Feynman lectures on
The fundamental theorem of calculus asserts the interrelated properties of integration and differentiation. It says that a function when differentiated, can be brought back by integrating (anti-derivative) or a function when integrated, can be brought back by differentiation.
First theorem: Let f be a function that is integrable on [a,x] for each x in [a,b], then let c be such that a≤c≤b and define a new function A as follows,
A(x)=∫_c^x▒f(x)dt, if a≤x≤b.
Then the derivative A’(x) exists at each point x in the open interval (a,b) where f is continuous, and for such x, A^' (x)=f(x).
This can be intuitively understood, by thinking of integration and differentiation as operations that work opposite to each …show more content…
According to the definition, if we progress to a certain term in sequence f(n), where it is sufficiently close to our limit L, then there difference of the two is less compared to a very small band of thickness ɛ. And when we surpass this certain value of N (which we define as an arbitrary threshold) to a greater value n, our difference becomes smaller and smaller until it is very close to or equal to zero. Therefore, very gallantly, we can say that at some point, our sequence will converge. And if it does not, then we can proceed to say that it diverges- that is the exact opposite, where the difference keeps getting bigger and …show more content…
We saw that the general term of our sequence was 1/n. So, as we keep increasing the value of n and eventually to infinity, the value of 1/n becomes zero. Thus, we fix that as our limit.
|f(n)-L|<ɛ, [L=0],
|f(n)|<ɛ,
Therefore, 1/n<ɛ, implies that, 1/ɛ<n.
We know that n≥N.
So, 1/ɛ<N.
Finally, we can claim that the sum of an infinite series is not just a distinct algebraic sum of the terms, but rather the “sum”, is a very special sum. It is obtained as the limit of the sequence of partial sums. The very concept of summation becomes hesitant when we progress further where the terms themselves become hazy for a clear operation. A ‘sum’ itself is a number and is just as it is, according to face value. The difference is how the sum manifests itself in a ‘series’, where we see whether it is converging or diverging and notice it moving across the plane, stopping and meandering.
Reference: T.M. Apostol: One variable Calculus. The Feynman lectures on