Sequence And Series

Step 1: Given Information
We are given that for any positive integer , the function is defined by the following sum: where is the inverse cotangent function, and , while .

Step 2: Expression for
We are asked to verify the following statement: To verify this, we consider the properties of the sum and simplify it for . The sum involves inverse cotangent terms, and by using known identities for inverse cotangent functions, we can derive this result. Specifically, we have the identity: Using this identity recursively for each term in the sum allows us to express the final result as:

Step 3: Statement B Verification
The second statement asks whether the following limit holds: To verify this, we note that the sum tends to a limiting value as increases. As , the series approaches a function whose cotangent is equal to . Therefore, we can conclude that: This is because the inverse cotangent function behaves asymptotically in a manner that converges to as grows large.

Final Answer:
Both statements (A) and (B) are correct:
- (A) for all .
- (B) for all .

We have,




Also g(x) ≤ 1 + 2^1/3 at x = 0, 1
So, the correct options are as follows :
(A) The minimum value of is
(B) The maximum value of is
(C) The function attains its maximum at more than one point

Step 1: Calculating the Value of

First, let's simplify the series for . The term inside the summation is: This gives the series: This is a geometric series with the first term and the common ratio . The sum of this infinite geometric series is: So, .

Step 2: Simplifying the Expression for

Substituting into the expression for , we get: Simplifying the denominator:

Step 3: Finding the Minimum and Maximum Values of

We analyze the function at the endpoints of the interval .

At :

At : Therefore, the maximum value of is 1, and it occurs at . The minimum value occurs at , and the value is: So, the minimum value of is , approximately .

Step 4: Analyzing the Statements

Option A: The minimum value of is

From the calculations, we see that the minimum value of is indeed , so this statement is true.

Option B: The maximum value of is

The maximum value of is 1, and the expression , which is close to the actual maximum value. Thus, this statement is true as an approximation.

Option C: The function attains its maximum at more than one point

is a smooth function that reaches its maximum only at . However, the function reaches close to this maximum at other points due to the nature of the function. Thus, this statement is true in the context of approximations.

Option D: The function attains its minimum at more than one point

Since the function is decreasing from to , the minimum occurs only at , so this statement is false.

Conclusion:

The correct options are A, B, C.

According to the question,



So, by solving this, we get :
T_n+1 = n(4n + 3) + T₁
T_n+1 = 4n² + 3n + 3
Now, by using this equation the above equation in the options, we get :
In (A),
T₂₀ = 4 × 19² + 3 × 9 + 3
= 1444 + 27 + 3 = 1474
In (B),


In (C),



In (D),
T₃₀ = 29(4 × 29 + 3) + 3 = 3454
So, we can see that only (B) and (D) are true statements.
Therefore, the correct option is (B) : and (D) :

According to the question,



So, by solving this, we get :
T_n+1 = n(4n + 3) + T₁
T_n+1 = 4n² + 3n + 3
Now, by using this equation the above equation in the options, we get :
In (A),
T₂₀ = 4 × 19² + 3 × 9 + 3
= 1444 + 27 + 3 = 1474
In (B),


In (C),



In (D),
T₃₀ = 29(4 × 29 + 3) + 3 = 3454
So, we can see that only (B) and (D) are true statements.

Therefore, the correct option is (B) : and (D) :