promath A sequence is called convergent if its limit is finite, and divergent if its limit is infinite (either +∞ or -∞). The difference between these two type of sequences will be discussed in this page. Convergent and Divergent Sequence Difference The differences of convergent and divergent sequences are listed in the table below. Convergent Squence … Read more
A divergent sequence is a sequence having limit either infinite (that is, does not converge to a specific finite limit). For example, the sequence {n} has infinite limit, hence a divergent sequence. Here we learn the definition and examples of divergent sequence. Divergent Sequence Definition A sequence {xn} is said be divergent if its limit … Read more
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