The On-Line Encyclopedia of Integer Sequences® (OEIS®)
Grandi's Series: 1-1+1-1+1... - YouTube
A sequence is an ordered list of infinite expressions.
A series a sum of infinite expressions.
When calculating the sum of an series, we look at the sequence of partial sums, defined to be the sum up to the n-th terms. If the limit of the partial sums exists and is bound, we assign the limit as the sum of the series.
We can add, subtract terms by terms of series' by adding and subtracting their values.
Alternatively you can tweak the definition of series, say looking at the sequence of the averages of partial sums, if the limit exists (as in Grandi's Series), it is said to be Cesaro convergent. This trick can be applied repeatedly, look at the sequence of averages of the averages of partial sums. Then some of the previously unbound series are now Cesaro convergent so values can be assigned to it.
When the limit is unbound, the sum of the series is undefined. Applying Riemann Zeta function for s <= 1 with analytic continuation is actually redefining the function for s <= 1. We do not have to make the redefinition to be symmetrical to the s > 1 portion (this is chosen to make the redefinition derivable (analytic)), but if we do, we will assign -1/2 to zeta(-1) and such. It's not actually taking the limit, it's just a way to assign value to the symbol zeta(-1).
Like we (the human race) did to extend rational numbers to real numbers, then to extend real numbers to complex numbers, all of these series will make sense if we relax the axioms.
Fourier Series/Fourier Transform
The Fourier Series and Fourier Transform Demystified - YouTube
But what is the Fourier Transform? A visual introduction. - YouTube
The Most Important Algorithm Of All Time - YouTube FFT
傅立叶变换如何理解？美颜和变声都是什么原理？李永乐老师告诉你 - YouTube
Sum of all natural numbers
This sum of diverging series (ζ(-1) = -1/12) is applied in many physical fields.
1 + 2 + 3 + 4 + … - Wikiwand
自然数之和等于-1/12 有何含义？隐藏在卡西米尔效应中的无穷大 - YouTube
Why -1/12 is a gold nugget - YouTube
Numberphile v. Math: the truth about 1+2+3+...=-1/12 - YouTube
Response to Numberphile's ASTOUNDING 1+2+3+4+... = minus 1/12 (sum of natural numbers to infinity) - YouTube
Ramanujan: Making sense of 1+2+3+... = -1/12 and Co. - YouTube
The Riemann Hypothesis, Explained - YouTube
The sum of all counting numbers equals WHAT? - YouTube
1+2+3+4+...=-1/12？李永乐老师讲黎曼猜想（1） - YouTube
所有自然數之和是-1/12？它在物理學中還有特別的應用？
所有自然数之和是-1/12？它在物理学中还有特别的应用？-返朴-财新博客-财新网
Riemann Hypothesis
ζ(s) = 1/1^s+1/2^s+...
Riemann hypothesis - Wikiwand
s > 1 originally, Riemann ζ function analytically extended the function to complex plane and dropped this restriction
Visualizing the Riemann zeta function and analytic continuation - YouTube
The Riemann Hypothesis, Explained - YouTube
Riemann's paradox: pi = infinity minus infinity - YouTube Riemann Rearrangement Theorem
Riemann Hypothesis - Numberphile - YouTube
Visualizing the Riemann zeta function and analytic continuation - YouTube
质数有多重要？数学家欧拉和高斯是如何研究质数的 ？李永乐老师讲黎曼猜想（2） - YouTube
悬赏 100 万美元的“黎曼猜想”有多难？李永乐老师讲什么是黎曼猜想（3） - YouTube
你的生日是唯一的嗎？小學生也能聽得懂的數學之黎曼猜想（一）| 雅桑了嗎 - YouTube
最難掙的 100 萬美元！證明黎曼猜想是錯的反而更偉大？（二） | 雅桑了嗎 - YouTube
素数（二）黎曼猜想和素数有啥关系？欧拉发现黎曼猜想的金钥匙！ - YouTube
素数（三）黎曼猜想是什么？它被证明了吗？十分钟看懂黎曼猜想与解析延拓 - YouTube
素数（四）黎曼猜想中的非平凡零点和素数的分布规律有什么关系？ - YouTube