# INFINITE SERIES

### BINOMIAL THEOREM FOR ANY INDEX

If | x | < 1, then for any n The following are some important properties of this expansion:
1. In this expansions there are infinitely many terms.
2. This expansion is valid for | x | < 1 and first term unity.
3. The general term of this expansion is 4. When x is small compared with 1, we see that the terms finally get smaller and smaller. If x is very small compared with 1, we take 1 as a first approximation to the value of (1 + x)n or 1 + nx as a second approximation.
5. Replacing n by –n in the above expansion, we get Replacing x by –x in this expansion, we get Hence,       ### EXPONENTIALS

The number e
By the Binomial Theorem, if  n > 1, = If n is indefinitely large, then etc. Then the right hand side = Thus, the limiting value of when n is infinite, is the sum of the series The sum of this series is denoted by the letter 'e'. Hence, we have, RESULTS :
• e is an irrational number.
• 2 < e < 3. The approximate value of e is e = 2.71828.

### EXPONENTIAL SERIES

To prove that If n is natural number greater than 1, then for any x, or Taking limit as , we get  To expand ax in ascending powers of x
We have Let ec = a so that c = loge a.
Then ax = ec x = or This is called the exponential theorem.

SOME IMPORTANT FORMULAE
1. 2. 3. 4. 5. 6. 7. 8. Note :
(i) (ii) ### LOGARITHMS (DEFINITION)

For each positive real number , the unique real number m is called the logarithm of b to the base a written as logab, if am = b.
• Thus logab = m, if and only if am = b, 'log' being the abbreviation of the word 'logarithm'.
• The following properties of logarithm follow immediately from the definition :
loga 1 = 0 because a0 = 1
loga a = 1 because a1 = a
loga ax = x because ax = ax
• Moreover, as a is a positive real number, b is also a positive real number. Thus, we define the logarithm of only a positive real number to a positive real number base other than 1. Negative numbers and zero have no logarithms. The expressions log (–2), log 0, are meaningless.

### LAWS OF LOGARITHMS

If a > 0 and , then
1. 2. 3. 4. Change of base : For any positive real numbers r and b, ,
5. 6. 7. 8. for any positive x and c not equal to 1.
9. 10. 11. 12. ### LOGARITHMIC SERIES

If a is positive, then by the exponential theorem, we have
ay = 1 + y loge a + Putting a = 1 + x, we get If | x | < 1, then by Binomial theorem, we have = + terms involving higher powers of y
Thus,  This is an identity in y, therefore, equating coefficients of y on both sides we get This is known as the Logarithmic Series.

SOME IMPORTANT RESULTS
1.  2.  On Subtracting (2) from (1), we get
1. (–1 < x < 1)
2. ### CALCULATION OF LOGARITHMS

We have ...(1)
Suppose we have to find loge2. Putting , we get On putting in (1), we get
Similarly, we can find and so on.
Logarithms of numbers calculated to the base e are called NAPERIAN LOGARITHMS or NATURAL LOGARITHMS.
Logarithms to the base 10 are called COMMON LOGARITHMS. or  Hence, the logarithms of any number to the base 10 can be obtained by multiplying the logarithms of the number to the base e by 0.43429448.
Also we have 2 < e < 3
And logex is an increasing function, therefore log 2 < loge e < log 3  log 2 < 1 < log 3.

Example :
Prove that Solution :
We have,    ## Want to know more

Want to Know More
Please fill in the details below:

## Latest NEET Articles\$type=three\$c=3\$author=hide\$comment=hide\$rm=hide\$date=hide\$snippet=hide

Name

Alternating Current,60,Basic Maths,2,best books for iit jee,2,best coaching institute for iit,1,best coaching institute for iit jee preparation,1,best iit jee coaching delhi,1,best iit jee coaching in delhi,2,best study material for iit jee,4,Blog,62,books for jee preparation,1,books recommended by iit toppers,3,Capacitance,3,CBSE NEET,9,cbse neet 2019,3,CBSE NEET 2020,1,cbse neet nic,1,Centre of Mass,2,Chemistry,58,Class 12 Physics,15,coaching for jee advanced,1,coaching institute for iit jee,2,Collision,2,Current Electricity,4,Electromagnetic Induction,3,Electronics,1,Electrostatics,3,Energy,1,Fluid Mechanics,4,Gravitation,2,Heat,4,iit admission,1,iit advanced,1,iit coaching centre,3,iit coaching centre in delhi,2,iit coaching classes,2,iit coaching in delhi,1,iit coaching institute in delhi,1,iit entrance exam,1,iit entrance exam syllabus,2,iit exam pattern,2,iit jee,4,iit jee 2019,3,iit jee advanced,2,iit jee books,3,iit jee coaching,2,iit jee exam,3,iit jee exam 2019,1,iit jee exam pattern,3,iit jee institute,1,iit jee main 2019,2,iit jee mains,2,iit jee mains syllabus,2,iit jee material,1,iit jee online test,3,iit jee practice test,3,iit jee preparation,6,iit jee preparation in delhi,2,iit jee preparation time,1,iit jee preparation tips by toppers,2,iit jee question paper,1,iit jee study material,3,iit jee study materials,2,iit jee syllabus,2,iit jee syllabus 2019,2,iit jee test,3,iit preparation,2,iit preparation books,5,iit preparation time table,2,iit preparation tips,2,iit syllabus,2,iit test series,3,IITJEE,100,JEE Advanced,82,jee advanced exam,1,jee advanced exam pattern,1,jee advanced paper,1,JEE Books,1,JEE Coaching Delhi,3,jee exam,3,jee exam 2019,6,JEE Exam Pattern,2,jee exam pattern 2019,1,jee exam preparation,1,JEE Main,85,jee main 2019,4,JEE Main application form,1,jee main coaching,1,JEE Main eligibility criteria,3,jee main exam,1,jee main exam 2019,3,jee main online question paper,1,jee main online test,3,jee main registration,2,jee main syllabus,2,jee mains question bank,1,jee mains test papers,3,JEE Mock Test,2,jee notes,1,jee past papers,1,JEE Preparation,2,jee preparation in delhi,1,jee preparation material,4,JEE Study Material,1,jee syllabus,6,JEE Syllabus Chemistry,1,JEE Syllabus Maths,1,JEE Syllabus Physics,1,jee test series,3,Kinematics,1,Latest article,1,Latest Articles,61,Latest News,1,latest news about neet exam,1,Laws of Motion,2,Magnetic Effect of Current,3,Magnetism,3,Modern Physics,1,NCERT Solutions,15,neet,2,neet 2019,1,neet 2019 eligibility criteria,1,neet 2019 exam date,2,neet 2019 test series,2,NEET 2020 Eligibility Criteria,1,neet application form 2019 last date,1,Neet Biology Syllabus,1,Neet Books,3,neet eligibility criteria,3,neet exam 2019,7,neet exam application,1,neet exam date,1,neet exam details,1,neet exam pattern,6,neet exam pattern 2019,2,neet examination,1,neet mock test 2019,1,Neet Notes,3,Neet Online Application Form,3,neet online test,2,neet past papers,1,neet physics syllabus,1,neet practice test,2,NEET preparation books,1,neet qualification marks,1,NEET question paper 2019,1,neet question papers,1,neet registration,1,Neet Study Material,3,neet syllabus,6,neet syllabus 2019,5,NEET Syllabus 2020,1,neet syllabus chemistry,1,neet syllabus for biology,1,neet syllabus for physics,1,neet test series,1,neet ug 2019,2,news,2,online study material for iit jee,1,Optical Instruments,1,Physics,110,physics books for iit jee,1,Power,1,Practical Physics,1,Ray Optics,1,Rotational Motion,3,SHM,3,Simple Harmonic Motion,3,study materials for iit jee,1,Study Notes,110,study notes for iit jee,1,Thermodynamics,4,Units and Dimensions,1,Vectors,2,Wave Motion,3,Wave Optics,1,Work,1,
ltr
item
Best IIT JEE Coaching Institute in Delhi | NEET Coaching in East Delhi | NEET Coaching in West Delhi: Infinite Series | Mathematics Notes for IITJEE Main
Infinite Series | Mathematics Notes for IITJEE Main