Tips for Improving Students' Math Skills | MATH TIPS

Many Students think that math is tough because somehow they are weak in math.

In this blog i post some tips and technique to improve your math skills.But before I am going to discuss about some problems in our study life: (Must read the below section)

➡️Our Study Life:

Most of the students study only for marks and nothing else.This is not the fault of the students, this is family pressure.

  I mostly says about India. Look at some example of this.

Example:

               Father(Family):- Son just secure 90% above marks in your 10th board and after that there will be no pressure for study.

      ( After Securing 85% in 10th board)

Father(Family):- Shame* on you. You can't secure just 90% in your exam.What will you do in your future . You will beg for money from house door to house door. Even if no one will give his daughter to you.

And  your friend Rahul(Son of Ramlal Sharma) got 93% , learn something from him.

          In our days we had to walk 10Km to go to school. If we got the facility that you have , then we must be an IAS or IPS Officer.

 And a interesting fact that is The same story will repeat after 12th board,Engineering/Medical entrances,job entrances and will go on.

After hearing this some students think that Marks is everything .Then they start study only for marks and  memories everything without understanding.

So Marks is not everything. We All know that mark is important but understanding is more important than marks.

If You really Want to Develop your math skills you should go on. Or Just Leave from here.

➡️1.Understanding the Concepts:

You can memorize formulae and rules to complete many math problems, but this doesn't mean that you understand the underlying concepts behind what you're doing. This makes it harder to successfully solve problems, as well as making it nearly impossible to easily absorb new information. Taking the time to make sure you understand why you're doing what you're doing can help your math skills immensely.

➡️2.Find New Concepts and Practice Problems:

Jumping directly into solving problems can lead to frustration and confusion. Try to study your textbooks and pay attention in class. You should also work on any practice problems your teachers assign before completing any assignments. This gives you a chance to absorb what you're learning.

➡️3.Work On Additional Exercises:

Practice makes perfect, even with math. If you are struggling with a particular kind of problem, you can improve by working on solving additional problems. You can start out with simplified problems of the same type, and move up in difficulty as you become more comfortable with finding the solutions.

➡️4.Make math part of your life:

Take the time to apply math in common situations. For example, if, say, a sweater that’s regularly $38 is on sale for 30% off, what is the sale price? Or if you need to double a recipe that calls for 3/4 cup of flour, how much flour will you need?

➡️5.Play math games.

Math games are good tools for honing your math skills and are designed to let you have fun while doing it.

There are some math Games for Android and PC

• 2048.(Android)
• Math Games.(Android)
• Math Land.(Android)
• Khan Academy Kids.(Android)
• Math Master.(Android)
• DragonBox5+(PC)
• Prodigy(PC)
• Minecraft(PC/Android)

➡️6.Study From Different Sources:

Friendship is the Great resource.You may hire one of your good friend as your study partner.

   Once Dr. APJ Abdul Kalam Sir Said “One best book is equal to hundred good friends but one good friend is equal to a library.”

Internet is now the most cool and intelligent source of anything.You may google something about your theory. 

My recommended  websites are

• MIT open courseware
• Khan Academy
• TedEd(Youtube Channel)
• wolframalpha
• Wikiversiry

CONVERGENCE OF SEQUENCE AND SERIES (Part -1)| REAL ANALYSIS

INTRODUCTION:

All of you must read the 1st part of real analysis.If you are not then just CLICK HERE TO READ.

Though I repeat some basic knowledge about real analysis.

Real Sequence:
A real sequence is a map ∱: N ➝ R
If we write ∱(n)= x_n , n ∈ N 
then it is customary to denote the sequence
∱ by (x_n)_n=1 or sometimes (x₁ ,x₂ ,x₃ ,...,x_n )

The set {x_n : n ∈ N } is the range of the sequence.

A sequence in R is a list or ordered set: (a₁ , a₂ , a₃ , ... ) of real numbers.

Bounded Sequence:

The sequence  (x_n) is said to be bounded above if there exist a real number 'M' such that x_n ≤ M  for all n ∈ N .

       It is said to be bounded below if there exist m∈ R such that x_n ≥ m  for all n ∈ N.

A sequence is said to be bounded if it is bounded above and below.
So, simply (x_n) is bounded if and only if there exist a constant K > 0 such that

      |x_n| < K for n ∈ N
i.e;   -K < x_n < K

Convergence:

A sequence x_n  is called convergent if there exist a real number ' a ' such that for every є > 0 ,there exist n₀ ∈ N  depending upon є such that for n > n₀

                                         |x_n - a| < є

that is,

                               a - є < x_n < a+ є ,

Simply,

$$

Definition: We say x_n→∞ as n→∞ if for every M in R there is a natural number N so that x_n≥ M for all n≥N. We say x_n→−∞ as n→∞ if for every M in R there is a natural number N so that x_n≤ M for all n≥N.

Example: 1, 1/2, 1/4, 1/8, 1/16 ,.........

This Sequence converge to a a finite point so it is Convergence Sequence.

Divergence:

A sequence that is not converge is called Diverge.

Example: 1,2,3,4,......

In the above example the sequence leads to +∞ . So it is a Divergent Sequence.

Read the below Question Carefully for Better Understand.

QUESTION:

1. Check 1/2 , 2/3 , 3/4 ,4/5 ,5/6 ........   is Convergence or Divergence ?

ANSWER:

Step1: Find the n_th term

t_n = n/(n+1)

Step2: Find the limit of the n_th term with n↠ ∞

Similarly , if we don't get any finite number then it will be Divergent Sequence.

Limit Superior and Limit Interior Sequences:

Let x_n  be the sequence of real numbers , for any k, let S_k = { x_n : n > k} . Then

Also Limit Superior and Limit Interior is categories under Convergence and Divergence. Which we have already discussed above.

Other Types Of Sequences:

Monotonic Sequences:

A sequence of increasing(Diverging) or decreasing(Converging) is called Monotonic Sequences.

Example:

1,1/3,1/9 ,1/27 ,........ is a Converging Sequence.

0,2,4,6,8,........ is a Diverging Sequence.

Oscillatory Sequence:

The oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point.

Example: (-1) ⁿ  is an example of Oscillatory Sequence.

Cauchy's Sequence:

We say that a sequence of real numbers {a_n } is a Cauchy sequence provided that for every Є>0, there is a natural number N so that when n,m≥N, we have that|a_n -a_m |≤ Є.

Basic Properties of Cauchy's Sequence:

1. Every convergent sequence is a Cauchy sequence,
2. Every Cauchy sequence of real (or complex) numbers is bounded .
3. If in a metric space, a Cauchy sequence possessing a convergent sub sequence with limit is itself convergent and has the same limit.