Math by Tathagata
An alternating series is a series containing terms alternatively positive and negative. One can check its convergence using Leibnitz’s test. In this article, we will study alternating series, Leibnitz’s test with solved problems. Definition of Alternating Series A series of the form $\displaystyle \sum_{n=1}^\infty$ (-1)n+1an, where an>0 for all n, is called an alternating series. … Read more
Cauchy’s Root test for series is used to test the convergence or divergence of an infinite series. It states that if limn→∞ an1/n is less than 1 then the series ∑an converges and if the limit is greater than 1 then the series diverges. In this article, we will study Cauchy root test of a … Read more
The ratio test for series is useful to check a series is convergent or divergent. Here we consider the limit of the quotient of (n+1)-th and n-th terms of the series, and depending on this limit is greater or less than 1 we decide its convergence. In this article, we study the ratio test of … Read more
The comparison test for series is used to check whether the series is convergent or not. Comparison test is useful when one can compare the given series with another series whose convergence is known. In this article, we learn about comparison test of a series with its limit form and some solved examples. Comparison Test … Read more
For a sequence {an}, if we add all the members of that sequence, then we get an infinite series. Here, a1+a2+a3+… is called infinite series (simply, a series) generated by the sequence {an}. In this article, we will study the convergence of a series along with its definition, formula, examples. The above series is denoted … Read more
The Fourier series of mod x in [-π, π] is given by |x| = π/2 -4/π [cosx/12 + cos3x/32 + cos5x/52 + …]. In this article, we learn how to find the Fourier series of mod x. Fourier Series of |x| Question: Find the Fourier series representation of f(x) = |x| in -π<x<π. Answer: The … Read more
A half range Fourier series of a function f(x) is a Fourier series of f(x) in the interval [0, L] although the function is defined for [-L, L]. Here, we learn about half range Fourier series with examples. Previously, we have learnt about the Fourier series representation of a periodic function of period 2L or … Read more
The Fourier Series of even and odd functions have a nice description. Recall that the Fourier series expansion of a function f(x) with period 2π is given by f(x) = a0 + ∑n=1[ ancos(nx) + bn sin(nx) ] where a0, an, bn are the Fourier coefficients of f(x) (provided that the sum exists). In this article, we learn … Read more
The Fourier series of a periodic function with period P is determined based on the following properties of integrals: $\int_a^{a+P} f(x) ~ dx = \int_0^p f(x) ~dx$. This means the area under the curve y=f(x) over any interval of length P is always the same. Using this property, one can express a function of period … Read more
A Fourier series of a function f(x) with period 2π is an infinite trigonometric series given by f(x) = a0 + ∑n=1[ ancos(nx) + bn sin(nx) ] if it exists. The constants a0, an, bn are called Fourier coefficients of f(x). In this article, let us learn Fourier series along with its formula and examples. … Read more