Wrote a blog post on the Fourier Series!

Sequences are an infinite or finite bunch of terms that come one after another.

a sequence in $\mathbb{R}$ is a function $f:\mathbb{N} \to \mathbb{R}$. one could say that it can be represented as an "infinite list of terms" but it certainly is never a "finite bunch of terms"

Most sequences follow some kind of pattern

no matter how you want to try define the worst "most" this is never going to be true

we can think of sequences as a bunch of numbers strung together that follow a general formula to form a pattern

same as the previous point

Sequences can either be infinite (they go on forever) or finite (they have an end value).

sequences do not ever have an "end value" (see first point), unless you mean a limit, but this is unrelated (and requires discussion of convergence - but it should also be noted that the sequence can converge to a value without that value being any element of the sequence)

what you get when you add all the terms in sequence together.

if a sequence has infinitely many terms, you cannot "add them all together". a series is just a sequence, ie a function $f:\mathbb{N} \to \mathbb{R}$. at best you could say that the limit (assuming it converges) of a series can be thought of as "adding all the terms of a sequence together" but this is imprecise and does not lend itself to generalization.

we can see that it's an infinite series; it has no end

a series is still just a sequence in $\mathbb{R}$ (ie. a function $f:\mathbb{N} \to \mathbb{R}$ ) so as written above this still doesn't make sense

we can think of Series as the sum of all terms in finite sequence.

as written above, at best you can say that one can pretend the limit of a series is the "sum of all the terms in a sequence". definitely a series is not "the sum of all terms in a finite sequence" (as written above, 'finite sequence' is not defined, and if you want to call an n-tuple a 'finite sequence' then i would say that sum of all the numbers in an n-tuple is simply a number, not a "series". if our n-tuple is (1,2,3) then $1+2+3=6$, and $6$ is not a series, just a number)

I'll be calling him Barry for brevity's sake.

this isn't mathematically incorrect it's just very disrespectful. i get you're trying to be funny

any periodic function can be represented as a series of sines and cosines

this isn't true. "any" should be replaced with "some". there are very very precise conditions on when a function can be represented by a Fourier series

A function is known to be periodic with a period $T$ – only if $f(x+T) = f(x)$. That means, the output value repeats itself every finite number of steps.

this should not say "known to be" and "only if". this is the definition of being periodic. it should say something like "we define a function to be periodic if..." or "a function is called periodic if..."

/r/math Thread Parent