# Summation of series(method of differences),When we use #∑f(r)-f(r+1#),how can we know it is #f(1)-f(n+1)#, but not #f(n)-f(1+1)#? Also, how can we know it is #f(n+1)-f(1)# but not #f(1+1)-f(n)# when we use #∑f(r+1)-f(r)#?

# sum_(r=1)^n f(r)-f(r+1) = f(1)-f(n+1)#

# sum_(r=1)^n f(r+1)-f(r) = f(n+1) -f(1)#

Write out the terms and see what happens:

Consider:

And as shown almost all terms vanish, leaving:

Whereas with:

Similarly, almost all terms vanish, leaving:

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Refer to the Explanation.

Otherwise, by what we have proved above,

Enjoy Maths.!

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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