In calculus, the Taylor series is widely used to find the power series of the one variable functions with respect to a specific point. It can also take the order of the function that tells how many derivatives of the function must be involved in the series.

The derivative is a branch of calculus that differentiate the function according to the independent variable. In this post, we will learn the definition, formula, examples, and solutions of the Taylor series.

**What is the Taylor series?**

In calculus, a power series that gives the expansion of a function f(y) in the region of a point “**b**” provided that in a region the function is continuous and all the derivatives exist is known as a Taylor series.

It can also have defined as a function of the endless sum of the terms that are expressed in terms of the differential of a function at a center point. It is usually used to expand a function in series by taking the specific point and the nth order.

The function must be continuous and the derivative of that function must exist. If the differentiation of the function is not existing, then the series of Taylor does not exist for that function.

### moto g power (2022)

UP TO 3-DAY BATTERY

ENDLESS POSSIBILITIES.

Color Dark Grove or Ice Blue

Internal Storage 64GB or 128GB

Unlocked

Was US$199.99** Now US$189.99**

**Taylor series Formula**

Here is a general formula of the Taylor series:

**F(y) = f(b) + f’(b) (y – b) / 1! + f’’(b) (y – b) ^{2}/ 2! + f’’’(b) (y – b)^{3}/ 3! + … + f^{n}(b) (y – b)^{ n}/ n!**

The formula of the Taylor series in the form of summation is given below.

**f**is the n^{n}(b)^{th}differential of the function at a specific point**b**.**n**is the order of the differential or total numbers.**b**is the specific point.- F(y) is the series of the given function.

To get the result of the Taylor series according to the above formula, use a Taylor series calculator.

**How to solve the problems of the Taylor series?**

The problems of the Taylor series can be calculated easily with the help of its formula. For solving the problems of the Taylor series, the knowledge of finding derivatives is compulsory. Here are a few examples of the Taylor series.

**Example 1**

Evaluate the Taylor series of y * cos(y) around the center point 5 and the order is 6.

**Solution**

**Step 1:** First of all, take the given information into the problem.

f(y) = ycos(y)

order of the function= n = 5

center point = b = 2

**Step 2:** Now take the formula of the Taylor series and expand it according to the given data.

According to the given order, this equation becomes.

F(y) = f(b) + f’(b) (y – b)/1! + f’’(b) (y – b)^{2}/2! + f’’’(b) (y – b)^{3}/3! + f’’’’(b) (y – b)^{4}/4! + f’’’’’(b) (y – b)^{5}/5! + f’’’’’’(b) (y – b)^{6}/6! … (1)

**Step 3:** Now evaluate the first six derivatives of ycos(y) with respect to “y”

f(y) = ycos(y)

f’(y) = cos(y) – ysin(y)

f’’(y) = -sin(y) – (sin(y) + ycos(y)) = -2sin(y) – ycos(y)

f’’’(y) = -2cos(y) – cos(y) – (-ysin(y)) = -2cos(y) – cos(y) + ysin(y) = -3cosy + ysin(y)

f’’’’(y) = -3(-sin(y)) + sin(y) + ycos(y) = 3sin(y) + sin(y) + ycos(y) = 4sin(y) + ycos(y)

f’’’’’(y) = 4cos(y) + cos(y) + (-ysin(y)) = 5cos(y) – ysin(y)

f’’’’’’(y) = -5sin(y) – (sin(y) + ycos(y)) = -5sin(y) – sin(y) – ycos(y) = -6sin(y) – ycos(y)

**For n = 0**

f(b) (y – b)^{0}/ 0! = f(b) = bcos(b)

**For n = 1**

f’(b) (y – b)^{1}/ 1! = f’(b) (y – b) = (cos(b) – bsin(b)) (y – b)

**For n = 2**

f’’(b) (y – b)^{2}/ 2! = f’’(b) (y – b)^{2}/2 = ½ (-2sin(b) – bcos(b)) (y – b)^{2}

**For n = 3**

f’’’(b) (y – b)^{3}/3! = f’’’(b) (y – b)^{3}/6 = 1 /6 (-3cos(b) – bsin(b)) (y – b)^{3}

**For n = 4**

f’’’’(b) (y – b)^{4}/4! = f’’’’(b) (y – b)^{4}/24 = 1 /24 (4sin(b) + bcos(b)) (y – b)^{4}

**For n = 5**

f’’’’’(b) (y – b)^{5}/5! = f’’’’’(b) (y – b)^{5}/120 = 1 /120 (5cos(b) – bsin(b)) (y – b)^{5}

**For n = 6**

f’’’’’’(b) (y – b)^{6}/6! = f’’’’’’(b) (y – b)^{6}/720 = 1 /720 (-6sin(b) – bcos(b)) (y – b)^{6}

**Step 5:** Now Put the calculated values in (1)

F(y) = bcos(b) + (cos(b) – bsin(b)) (y – b) + ½ (-2sin(b) – bcos(b)) (y – b)^{2 }+ 1 /6 (-3cos(b) – bsin(b)) (y – b)^{3 }+ 1 /24 (4sin(b) + bcos(b)) (y – b)^{4 }+ 1 /120 (5cos(b) – bsin(b)) (y – b)^{5} + 1 /720 (-6sin(b) – bcos(b)) (y – b)^{6}

**Step 6:** Now replace b = 5 in the above series.

F(y) = 5cos(5) + (cos(5) – 5sin(5)) (y – 5) + ½ (-2sin(5) – 5cos(5)) (y – 5)^{2 }+ 1 /6 (-3cos(5) – 5sin(5)) (y – 5)^{3 }+ 1 /24 (4sin(5) + 5cos(5)) (y – 5)^{4 }+ 1 /120 (5cos(5) – 5sin(5)) (y – 5)^{5} + 1 /720 (-6sin(5) – 5cos(5)) (y – 5)^{6}

This is the required expansion of the cos(x) by using the Taylor series

**Example 2**

Evaluate the Taylor series of y^4 around the center point 20 and the order is 5.

**Solution**

**Step 1:** First of all, take the given information into the problem.

f(y) = y^4

order of the function= n = 5

center point = b = 20

**Step 2:** Now take the formula of the Taylor series and expand it according to the given data.

According to the given order, this equation becomes.

F(y) = f(b) + f’(b) (y – b)/1! + f’’(b) (y – b)^{2}/2! + f’’’(b) (y – b)^{3}/3! + f’’’’(b) (y – b)^{4}/4! + f’’’’’(b) (y – b)^{5}/5! … (1)

**Step 3:** Now evaluate the first six derivatives of y^4 with respect to “y”

f(y) = y^4

f’(y) = 4y^3

f’’(y) = 12y^2

f’’’(y) = 24y

f’’’’(y) = 24

f’’’’’(y) = 0

**For n = 0**

f(b) (y – b)^{0}/ 0! = f(b) = y^4 = b^4

**For n = 1**

f’(b) (y – b)^{1}/ 1! = f’(b) (y – b) = 4y^3 (y – b) = 4b^3 (y – b)

**For n = 2**

f’’(b) (y – b)^{2}/ 2! = f’’(b) (y – b)^{2}/2 = ½ 12y^2 (y – b)^{2} = 12b^2 (y – b)^{2}

**For n = 3**

f’’’(b) (y – b)^{3}/3! = f’’’(b) (y – b)^{3}/6 = 1 /6 24y (y – b)^{3} = 4b (y – b)^{3}

**For n = 4**

f’’’’(b) (y – b)^{4}/4! = f’’’’(b) (y – b)^{4}/24 = 1 /24 (24) (y – b)^{4} = (y – b)^{4}

**For n = 5**

f’’’’’(b) (y – b)^{5}/5! = f’’’’’(b) (y – b)^{5}/120 = 1/120 (0) (y – b)^{5} = 0

**Step 5:** Now Put the calculated values in (1)

F(y) = b^4 + 4b^3 (y – b) + 12b^2 (y – b)^{2 }+ 4b (y – b)^{3} + (y – b)^{4} + 0

**Step 6:** Now replace b = 2 in the above series.

F(y) = 2^4 + 4(2^3) (y – 2) + 12(2^2) (y – 2)^{2 }+ 4(2) (y – 2)^{3} + (y – 2)^{4} + 0

F(y) = 16 + 4(8) (y – 2) + 12(4) (y – 2)^{2 }+ 4(2) (y – 2)^{3} + (y – 2)^{4}

F(y) = 16 + 24 (y – 2) + 48 (y – 2)^{2 }+ 8 (y – 2)^{3} + (y – 2)^{4}

## Conclusion

In this post, we have covered all the basics of the Taylor series along with definition, formula, and solved examples. Now the confusion about the Taylor series can be cleared by learning this post.

### Every month in 2023 we will be giving away one *PlayStation 5*. To qualify join our *Facebook* **group**, *TikTok* and Subscribe to our **Sweet TnT Magazine YouTube channel**

*PlayStation 5*

**group**

*TikTok*

**Sweet TnT Magazine YouTube channel**

### Apple – iPhone 14 Pro Max 256GB (Unlocked, US version)

~~US$1,359.99~~ **US$899.99**

When you buy something through our retail links, we may earn commission and the retailer may receive certain auditable data for accounting purposes.

_____________________________

**You may also like:**

**Top 5 online calculators which are useful for learning math**

**6 benefits of using technology in math class**

**Digital transformation market worth US$1247.5 billion by 2026**

**Programming languages you should learn for robotics in 2022**

**This is why you should choose a tech career**

**Studying abroad: 12 useful tips for students**

**Harvard University: Access over 100 free online courses**

**Improve your intelligence daily**

**Everything you need to know about running a private fund**

### Meta Quest 2 (Oculus) – Advanced All-In-One Virtual Reality Headset – 128GB

Beat Saber included when you buy Meta Quest 2

US$395.00

You must log in to post a comment.