Grandi's series | Divergent series

Summation of Grandi's series

The formal manipulations that lead to 1 − 1 + 1 − 1 + · · · being assigned a value of 1⁄2 include: * Adding or subtracting two series term-by-term, * Multiplying through by a scalar term-by-term, * "Shifting" the series with no change in the sum, and * Increasing the sum by adding a new term to the series' head. These are all legal manipulations for sums of convergent series, but 1 − 1 + 1 − 1 + · · · is not a convergent series. Nonetheless, there are many summation methods that respect these manipulations and that do assign a "sum" to Grandi's series. Two of the simplest methods are Cesàro summation and Abel summation. (Wikipedia).

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I Just Can't Believe what the Math is Telling Me

The Grandi's Series history is quite famous. The sum 1-1+1-1+1-1 to infinity should clearly diverge with no dispute. However, this 1-1+1-1+1-1 infinite series has a very controversial answer: 1-1+1-1 = 1/2 ?! How could this be? Here's my option on the Grandi's Series answer and a Grandi'

From playlist Series, Series, Series

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1-1+1-1+... Grandi series from a complex analysis angle

In this video I talk about switching the limits of an expression that amounts to the Grandi series - a series which is divergent in real and comlex analysis. I'm also ranting a bit on the History of Science and math StackExchange page and other communities.

From playlist Programming

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Finding the sum or an arithmetic series using summation notation

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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Learn to use summation notation for an arithmetic series to find the sum

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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Ramanujan Summation

The third video in a series about Ramanujan.This one is about Ramanujan Summation. Here's the wikipedia page for further reading: https://en.wikipedia.org/wiki/Ramanujan_summation Euler-Maclaurin Formula https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula --------- Here is a

From playlist My Maths Videos

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Using sigma sum notation to evaluate the partial sum

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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What is the sum of an arithmetic series using the sum formula

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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This Video Will Open Your Mind

Head to https://squarespace.com/brithemathguy to save 10% off your first purchase of a website or domain using code BRITHEMATHGUY The 1-2+3-4+5-6 series is a bit a mathematical wonder. On one hand, this 1-2+3-4+5-6 series diverges. On the other hand, there are some techniques to assign in

From playlist Series, Series, Series

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Thoughts on the 1-1+1-1+... Series - A Gentle Discussion About Analytic Continuation

We're going to mess around with some divergent series in this video, namely the Grandi Series, 1-1+1-1+... and the related series 1-2+3-4+... and I'll hopefully convince you that the sums that are associated with these divergent series make some sense when seen through the lens of calculus

From playlist Math

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One minus one plus one minus one - Numberphile

Discussing the brain-bending Grandi's Series and Thomson's Lamp - featuring Dr James Grime. More links & stuff in full description below ↓↓↓ A little bit of extra footage from the very end of this interview at: http://youtu.be/TgGi1ls1X-0 (on Brady's own channel) See our Zeno's Paradox v

From playlist James Grime on Numberphile

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Given two terms find the sum of your arithmetic series

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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Grandi's Series: 1-1+1-1+1...

This is the first half of a lesson, watch the second half here: http://youtu.be/P913qwtXihk

From playlist Mathematical Exploration

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The finite part of infinity (ONLINE) by Joseph Samuel

Vigyan Adda The finite part of infinity (ONLINE) Speaker: Joseph Samuel (RRI & ICTS-TIFR, Bengaluru) When: 4:30 pm to 6:00 pm Sunday, 24 October 2021 Where: Livestream via the ICTS YouTube channel Abstract: - Ramanujan's notebooks contain the equation 1+2+3....= - 1/12. While this see

From playlist Vigyan Adda

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Solving the Basel Problem ... With a Divergent Series!

The Basel problem asks for the sum of the reciprocals of the square numbers, ζ(2), and was famously solved by Euler. In this video, we will find that a simple argument can be given for the solution of the Basel problem using a mysterious divergent series, which upon some simple operations

From playlist Math

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Paleo Profile - Dimetrodon

Today we examine this amazing, spine-backed Mammal-like Reptile or Synapsid! We will try to answer questions like: Did dimetrodon have fur? How did dimetrodon walk? And what is the deal with dimetrodon's back? Sources: http://www.bioone.org/doi/abs/10.3158/2158-5520-5.1.104 https://en.wik

From playlist Paleontology Videos

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The sum of all counting numbers equals WHAT?

This is the second half of a lesson, watch the first half here: http://youtu.be/yudhkUUzAgY This is a well-known and hugely controversial result. The "proof" I've demonstrated is not the only way to show it - there are far more sophisticated and convincing ways to do it - but suffice to s

From playlist Mathematical Exploration

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Math tutorial for determining the sum of an arithmetic series

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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Hilbert space | Borel summation | Geometric series | Formal power series | Spectral asymmetry | Abel's theorem | Mean value theorem | Ernesto Cesàro | Cesàro summation | Operator (physics) | Cauchy product