Sum of a Taylor Series

In this video, I showed how to find the sum of a given taylor series using the characteristics of known series

[stigastondogg]

Tip for people new to the Taylor series world

Remember your expansion for e^x
sin(x) is the imaginary part of e^ix
Substitute ix into expansion for e^x. Write out a few terms and you’ll be able to derive the summation expression for sin(x). The pattern that emerges will jog your memory and you’ll recognise it quickly.
Take the derivative and you get cos(x).

Bit less that you have to memorise 👍👍

[kingbeauregard]

Now that I've seen the solution, it's obvious in retrospect. Well done!

[barbaral]

this explanation made me smile. brilliant, thank you!

[briqel]

Thank you so much you explain very well I finally understand 😊

[elizabethnjason]

Thank you Sir for this, very helpful.

[nicholasroberts]

very nice question, thank you for posting!

[МаксимАндреев-щб]

pi-pi^3/(4^2•3!)+… = 4 (pi/4 - (pi/4)^3 / 3! + (pi/4)^5 / 5! - …) = 4sin(pi/4) = 4 sqrt(2)/2 = 2sqrt(2)

[holyshit]

This reminds me series expansion of sine

[kingbeauregard]

You what confounds me? Let's look at the terms of the Taylor series for sin (2*pi). All those terms - each of them (2*pi) raised to an integer exponent - somehow add up to exactly zero. It should not be possible, and yet it happens.