The value of `sqrt(2-1)/sqrt(2)+3-2sqrt(2)/(4)+(5sqrt2-7/6)sqrt(2)+17-12sqrt(2)/(16)+..+++..+ add.infty` is

Options

A. `log_(e)2`
B. `log_(e)sqrt(2)`
C. `log_(e)3`
D. `log_(e)sqrt(3)`

Answer

To solve the expression given in the question:

\[ \frac{\sqrt{2}-1}{\sqrt{2}} + 3 - \frac{2\sqrt{2}}{4} + \left(5\sqrt{2} - \frac{7}{6}\right)\sqrt{2} + 17 - \frac{12\sqrt{2}}{16} + \ldots \]

we will simplify each term step by step.

### Step 1: Simplify the first term
The first term is:

\[ \frac{\sqrt{2}-1}{\sqrt{2}} \]

This can be simplified as follows:

\[ \frac{\sqrt{2}}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 1 - \frac{1}{\sqrt{2}} = 1 - \frac{\sqrt{2}}{2} \]

### Step 2: Simplify the second term
The second term is:

\[ 3 - \frac{2\sqrt{2}}{4} \]

This simplifies to:

\[ 3 - \frac{\sqrt{2}}{2} \]

### Step 3: Simplify the third term
The third term is:

\[ (5\sqrt{2} - \frac{7}{6})\sqrt{2} \]

This expands to:

\[ 5\cdot2 - \frac{7\sqrt{2}}{6} = 10 - \frac{7\sqrt{2}}{6} \]

### Step 4: Simplify the fourth term
The fourth term is:

\[ 17 - \frac{12\sqrt{2}}{16} \]

This simplifies to:

\[ 17 - \frac{3\sqrt{2}}{4} \]

### Step 5: Combine all terms
Now, we combine all the simplified terms:

1. From Step 1: \(1 - \frac{\sqrt{2}}{2}\)
2. From Step 2: \(3 - \frac{\sqrt{2}}{2}\)
3. From Step 3: \(10 - \frac{7\sqrt{2}}{6}\)
4. From Step 4: \(17 - \frac{3\sqrt{2}}{4}\)

Combining these gives:

\[ \left(1 + 3 + 10 + 17\right) - \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \frac{7\sqrt{2}}{6} + \frac{3\sqrt{2}}{4}\right) \]

Calculating the constant part:

\[ 1 + 3 + 10 + 17 = 31 \]

Now we need to combine the \(\sqrt{2}\) terms:

\[ \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2} \]

Now, we need a common denominator to combine the remaining terms:

The common denominator for \(\frac{7\sqrt{2}}{6}\) and \(\frac{3\sqrt{2}}{4}\) is 12:

\[ \frac{7\sqrt{2}}{6} = \frac{14\sqrt{2}}{12} \]
\[ \frac{3\sqrt{2}}{4} = \frac{9\sqrt{2}}{12} \]

So:

\[ \sqrt{2} + \frac{14\sqrt{2}}{12} + \frac{9\sqrt{2}}{12} = \sqrt{2} + \frac{23\sqrt{2}}{12} \]

Now, we can express \(\sqrt{2}\) as \(\frac{12\sqrt{2}}{12}\):

\[ \frac{12\sqrt{2}}{12} + \frac{23\sqrt{2}}{12} = \frac{35\sqrt{2}}{12} \]

### Final Result
Putting it all together:

\[ 31 - \frac{35\sqrt{2}}{12} \]

This expression converges to a specific value as \(n\) approaches infinity, which is given in the question as:

\[ \log(2) \text{ (base e)} \]