In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.
Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.
Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π.
Mathematicians from the Kerala school were studying infinite series c. 1350 CE.
In the 17th century, James Gregory worked in the new decimal system on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of hypergeometric series and q-series.
https://en.wikipedia.org/wiki/Series_(mathematics)
Infinite series results from us wanting to know how the sum of a series behaves when the terms are infinitely many. We can write any series we see in its infinite series form, so it’s no surprise that infinite series has physics, biology, and engineering.
Infinite series represents the successive sum of a sequence of an infinite number of terms that are related to each other based on a given pattern or relation.
Isn’t it amazing how, through the advancement of mathematics, it is now possible for us to predict the sum of a series made of an endless number of terms?
What is an infinite series?
As our introduction says, infinite series represents the sum of the infinite number of terms formed by a sequence. Below are examples of infinite series:
1/2 + 1/4 + 1/6 + 1/8 + 1/10 +…
- This is an example of an infinite harmonic series, where the denominator increases by 2 as the series increase.
3+9+27+81+243+…
- This is an example of an infinite geometric series, where the next term is determined by multiplying 3 to the previous term.
These examples give us an idea of what makes up an infinite series, so let’s go ahead and formally define infinite series. In the next section, we’ll learn how we can express them in terms of sigma notation.
Infinite series definition
Let’s say, we have a finite sequence that consists terms of {𝑎~1~,𝑎~2~,…,𝑎~𝑛−1~,𝑎~𝑛~}, so the sum of its finite series can be expressed as 𝑎~1~ +𝑎~2~ +… +𝑎~𝑛−1~ +𝑎~𝑛~.
The only difference for infinite series, the terms extend beyond 𝑎~𝑛~, so the infinite series will be of the form 𝑎~1~ +𝑎~2~ +𝑎~3~ +…
or
How to find the sum of an infinite series?
At first, it may feel counter-intuitive to think that we can predict the sum of an infinite series. But thanks to limits and calculus, we’re able to create a systematic process to find the sum of a given infinite series.
But first, let’s take a look at this visual representation of an infinite geometric series.
This is a good example of how we can find the sum of infinite series. That’s because as we continue to add more terms (so take half of the previous area), we’ll see that when combined altogether, the total area of the shaded region will fill up almost the entire square’s region.
Any guess on the sum of the infinite series,
then? Visually, since the regions will eventually make up the entire square, the sum of the infinite series is 1.
But how do we confirm this mathematically? Before we dive right into the process of determining the sum of infinite series, let’s find out how to find the sum of a certain portion from a given infinite series.
How to find partial sum of infinite series?
The partial sum of an infinite series is simply the sum of a certain number of terms from the series. For example, the series 1/2 +1/4 + 1/8 is simply a part of the infinite series 1/2 + 1/4 + 1/8 + ...
This means that the partial sum of the first three terms of the infinite series shown above is equal to 1/2 + 1/4 + 1/8 = 7/8
How to find the infinite series’ sum based on its partial sum?
You might be wondering why we’re talking about partial sums when we’re supposedly dealing with the sums of infinite series. That’s because when we want to find the sum of an infinite series, we’ll need the expression of its partial sum.
Let’s say we have an infinite series,
, so its partial sum for the first 𝑛 terms will be
If the partial sum, 𝑆~𝑛~, converges, the infinite series, 𝑆, is expected to converge as well. In fact, lim 𝑛→∞ 𝑆~𝑛~ will represent the sum of the infinite series.
If the partial sum, 𝑆~𝑛~, diverges, the infinite series, 𝑆, is expected to diverge as well. In fact, it will not be possible for us to predict the sum of the series when the partial sum diverges.
Why don’t we go ahead and observe the following geometric series and see what happens with their partial sum and infinite series’s sum?
Starting with the series, 1/3 + 1/9 + 1/81 +…, we can see that the common ratio is 1/3 and the next terms will be smaller and will be approaching 0.
The partial sum of the series of 𝑛 terms will be equal to
where 𝑎 = 1/3 and 𝑟 = 1/3.
Let’s take a look at the limit of 𝑆~𝑛~ as 𝑛 approaches infinity.
Since the series converges to 1/2, the sum of the series is equal to 1/2.
What happens when the common ratio is greater than 1? Let’s see how the series, 2 +4 +8 +16 +… behaves to answer that question.
This times, we have 𝑟 =2 and 𝑎 =2.
Conceptually, we’re expecting the series to diverge, and that’s because as we add more terms, the partial sum drastically increases as well. We can confirm this guess by taking the limit of 𝑆~𝑛~ as it approaches infinity.
Since lim𝑛→∞ 𝑆𝑛 =∞, the infinite series’ diverges and will not have a fixed value.
Noticed how when the terms increase throughout the infinite series, the series diverges? That’s a helpful observation and something we need to keep in mind each time.
An important condition for the infinite series,
, to be convergent, lim𝑛→∞ 𝑎~𝑛~ must be equal to 0. This means that terms have to become smaller as the terms progress for the infinite series to be convergent.
https://www.storyofmathematics.com/infinite-series/
Yuktibhāṣā (Malayalam: യുക്തിഭാഷ, lit. 'Rationale'), [...] is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school. It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original.
Front and back cover of the Palm-leaf manuscripts of the Yuktibhasa, composed by Jyesthadeva in 1530
The work contains proofs and derivations of the theorems that it presents. Modern historians used to assert, based on the works of Indian mathematics that first became available, that early Indian scholars in astronomy and computation lacked in proofs, but Yuktibhāṣā demonstrates otherwise.
Some of its important topics include the infinite series expansions of functions; power series, including of π and π/4; trigonometric series of sine, cosine, and arctangent; Taylor series, including second and third order approximations of sine and cosine; radii, diameters and circumferences.
Yuktibhāṣā mainly gives rationale for the results in Nilakantha's Tantra Samgraha. It is considered an early text to give some ideas related to calculus like Taylor and infinite series of some trigonometric functions, predating Newton and Leibniz by two centuries. however they did not combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the powerful problem-solving tool we have today. The treatise was largely unnoticed outside India, as it was written in the local language of Malayalam. In modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work. For example, both Oxford University and the Royal Society of Great Britain have given attribution to pioneering mathematical theorems of Indian origin that predate their Western counterparts.
Yuktibhāṣā contains most of the developments of the earlier Kerala school, particularly Madhava and Nilakantha. The text is divided into two parts – the former deals with mathematical analysis and the latter with astronomy. Beyond this, the continuous text does not have any further division into subjects or topics, so published editions divide the work into chapters based on editorial judgment.
Pages from the Yuktibhasa
The first four chapters of the contain elementary mathematics, such as division, the Pythagorean theorem, square roots, etc. Novel ideas are not discussed until the sixth chapter on circumference of a circle. Yuktibhāṣā contains a derivation and proof for the power series of inverse tangent, discovered by Madhava. In the text, Jyesthadeva describes Madhava's series in the following manner:
The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.
The text also contains Madhava's infinite series expansion of π which he obtained from the expansion of the arc-tangent function.
Using a rational approximation of this series, he gave values of the number π as 3.14159265359, correct to 11 decimals, and as 3.1415926535898, correct to 13 decimals.
The text describes two methods for computing the value of π. First, obtain a rapidly converging series by transforming the original infinite series of π. By doing so, the first 21 terms of the infinite series
The text describes two methods for computing the value of π. First, obtain a rapidly converging series by transforming the original infinite series of π. By doing so, the first 21 terms of the infinite series
https://en.wikipedia.org/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81
Madhava (Born 1350 Died 1425 )was a mathematician from South India. He made some important advances in infinite series including finding the expansions for trigonometric functions.
All the mathematical writings of Madhava have been lost, although some of his texts on astronomy have survived. However his brilliant work in mathematics has been largely discovered by the reports of other Keralese mathematicians such as Nilakantha who lived about 100 years later.
Madhava discovered the series equivalent to the Maclaurin expansions of sin x, cos x, and arctanx around 1400, which is over two hundred years before they were rediscovered in Europe. Details appear in a number of works written by his followers such as Mahajyanayana prakara which means Method of computing the great sines. In fact this work had been claimed by some historians such as Sarma to be by Madhava himself but this seems highly unlikely and it is now accepted by most historians to be a 16th century work by a follower of Madhava.