promath A convergent sequence is a sequence having finite limit. For example, {1/n} is a convergent sequence with limit 0. Here, we study the definition and examples of convergent sequence. Convergent Sequence Definition A sequence {xn} is said be convergent if it has a finite limit L. That is, for every ε>0, there is a natural … Read more
An unbounded sequence is a sequence which is not bounded. For example the sequence {n} of natural numbers is unbounded. In this article, we study unbounded sequences with its definition and examples. Unbounded Sequence Definition A sequence {an} is said to be unbounded if there is no real number M such that |an| ≤ M. … Read more
No, a continuous function is not always differentiable. For example, f(x)=|x| is continuous but not differentiable at x=0. In this post, we will study the following: is a continuous function always differentiable? A continuous function may not be differentiable always. Although, a differentiable function is always continuous. Lets prove this fact here. Recall the definitions … Read more
The formula of 1-cos2x is given by 1-cos2x=2sin2x. In this post, we establish 1-cos2x formula and identity with some solved examples. Formula of 1-cos2x Answer: The 1-cos2x formula is 1-cos2x=2sin2x. Proof: Using the formula cos2θ= cos2θ -sin2θ, the given expression can be written as follows. 1-cos2x = 1- (cos2x -sin2x) = (1- cos2x) -sin2x = … Read more
In this article, we prove that a convergent sequence is bounded. A sequence is said to be convergent if it has finite limit. Question: Prove that A Convergent Sequence is Bounded. Proof Let us assume that {xn} is a convergent sequence. So it has finite limit, say L. That is, limn→∞ xn = L. So … Read more
A sequence {xn} is called a bounded sequence if k ≤ xn ≤ K for all natural numbers n. For example, {1/n} is a bounded sequence since 0 < 1/n ≤ 1 for all n. Let us now learn about bounded sequence. Bounded Sequence Definition A sequence {xn} is said to be bounded if k … Read more
The nth derivative of cosx is equal to cos(nπ/2 +x). The nth derivative of cos x is denoted by dn/dxn (cosx), and its formula is given as follows: $\boxed{\dfrac{d^n}{dx^n}\left( \cos x\right)=\cos \left(\dfrac{n \pi}{2}+x \right)}$ nth Derivative of cos x Question: Find the nth derivative of cosx. Answer: To find the nth derivative of cosx with … Read more
The nth derivative of sinx is equal to sin(nπ/2 +x). The nth derivative of sin x is denoted by dn/dxn (sinx), and its formula is given as follows: $\boxed{\dfrac{d^n}{dx^n}\left( \sin x\right)=\sin \left(\dfrac{n \pi}{2}+x \right)}$ nth Derivative of sin x Question: Find the nth Derivative of sinx. Answer: To find the nth derivative of sinx with … Read more
The nth derivative of 1/(ax+b) is equal to (-1)nn!an/(ax+b)n+1. The nth derivative of 1/(ax+b) is denoted by $\dfrac{d^n}{dx^n}\left( \dfrac{1}{ax+b}\right)$ and its formula is given below: $\boxed{\dfrac{d^n}{dx^n}\left( \dfrac{1}{ax+b}\right)=\dfrac{(-1)^n n! a^n}{(ax+b)^{n+1}}}$ nth Derivative of 1/(ax+b) Question: What is the nth Derivative of $\dfrac{1}{ax+b}$? Answer: Let us put y = $\dfrac{1}{x+b}$ = (ax+b)-1. Using the power rule $\dfrac{d}{dx}\left( … Read more
The nth derivative of xn is equal to n!. The nth derivative of x^n is denoted by $\frac{d^n}{dx^n}\left( x^n\right)$, and its formula is given as follows: $\boxed{\dfrac{d^n}{dx^n}\left( x^n\right)=n!}$ nth Derivative of xn Question: Find nth Derivative of xn. Answer: The nth derivative of x to the power n is obtained by repeatedly using the power … Read more