## How do you know when to use the limit comparison test?

**1 Answer**

- The
**comparison test**requires an≤bn (or vice-versa) for all n and an,bn≥0 for all n. - The
**limit comparison test**does not require that an≤bn (or vice-versa) for all n. - So, if you have an≥0,bn>0 for all n, you define c=limn→∞anbn.
- If c is positive and finite then either both diverge or both converge.

## Why does the limit comparison test work?

In other words, in the **limit comparison test** you **do** not know whether your series converge/diverge, so using **limits** you find whether they both will diverge or converge. In the **comparison test**, you know whether on converges/diverges and using that knowledge, attempt to find whether the other converges or diverges.

## How do you know when to use the alternating series test?

**You can say that an alternating series converges if two conditions are met:**

- Its nth term converges to zero.
- Its terms are non-increasing — in other words, each term is either smaller than or the same as its predecessor (ignoring the minus
**signs**).

## What happens if the limit comparison test equals 0?

**If the limit** is positive, then the terms are growing at the same rate, so both series converge or diverge together. **If the limit** is **zero**, then the bottom terms are growing more quickly than the top terms. Thus, **if** the bottom series converges, the top series, which is growing more slowly, must also converge.

## What is the limit test?

“**Limit test** is defined as quantitative or semi-quantitative **test** designed to identify and. control small quantities of impurity which is likely to be present in the substance. **Limit test** is generally carried out to determine the inorganic impurities present in the compound”.

## How do you know if a limit converges or diverges?

**If** limn→∞an lim n → ∞ exists and is finite we say that the **sequence** is **convergent**. **If** limn→∞an lim n → ∞ doesn’t exist or is infinite we say the **sequence diverges**.

## How do you prove a limit diverges?

To show divergence we must show that the **sequence** satisﬁes the negation of the deﬁnition of convergence. That is, we must show that for every r∈R there is an ε>0 such that for every N∈R, there is an n>N with |n−r|≥ε.

## When can the integral test be used?

The **Integral Test**

If you **can** define f so that it is a continuous, positive, decreasing function from 1 to infinity (including 1) such that a[n]=f(n), then the sum will converge if and only if the **integral** of f from 1 to infinity converges.

## What happens if alternating series test fails?

(2) “**If** a given **alternating series fails** to satisfy one or more of the above three conditions, then the **series** diverges.” We need to realize the basic logic here: The contraposition of “**If** A is true, then B is true.” is “**If** B is false, then A is false.” These two statements are equivalent.

## Does 1 sqrt converge?

Hence by the Integral Test sum **1**/**sqrt**(**n**) diverges. Hence, you cannot tell from the calculator whether it **converges** or diverges. sum **1**/**n** and the integral test gives: Hence the harmonic series diverges.

## How do you tell if an integral is proper or improper?

An **improper integral** is a definite **integral** that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. **Improper integrals** cannot be computed using a normal Riemann **integral**.

## What is the comparison theorem in calculus?

If f(x)≥g(x)≥0 f ( x ) ≥ g ( x ) ≥ 0 on the interval [a,∞) then, If ∫∞af(x)dx ∫ a ∞ f ( x ) d x converges then so does ∫∞ag(x)dx ∫ a ∞ g ( x ) d x. If ∫∞ag(x)dx ∫ a ∞ g ( x ) d x diverges then so does ∫∞af(x)dx ∫ a ∞ f ( x ) d x.

## What is the comparison test in calculus?

In mathematics, the comparison test, sometimes called the **direct comparison test** to distinguish it from similar related tests (especially the **limit comparison test**), provides a way of deducing the convergence or divergence of an infinite series or an improper integral.