pandas.api.types.infer_dtype should be nan aware
#### Code Sample, a copy-pastable example if possible

```python
# Your code here
In [6]: s = pd.Series(['a', np.nan, 'c'])

In [7]: pd.api.types.infer_dtype(s)
Out[7]: 'mixed'
```
#### Problem description

`infer_dtype` gives `mixed` as the inferred dtype when all values are of the same type modulo `NaN`s. I think we should change this to be `NaN` aware so that downstream tools can take advantage of this inference when operating on pandas objects.

#### Expected Output

Ideally,

```python
In [8]: pd.api.types.infer_dtype(s.dropna())
Out[8]: 'string'
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: None
python: 3.6.1.final.0
python-bits: 64
OS: Linux
OS-release: 4.11.6-3-ARCH
machine: x86_64
processor:
byteorder: little
LC_ALL: None
LANG: en_US.utf8
LOCALE: en_US.UTF-8

pandas: 0.20.2
pytest: 3.0.7
pip: 9.0.1
setuptools: 27.2.0
Cython: None
numpy: 1.13.1
scipy: 0.19.1
xarray: None
IPython: 6.0.0
sphinx: None
patsy: None
dateutil: 2.6.0
pytz: 2017.2
blosc: None
bottleneck: None
tables: None
numexpr: None
feather: None
matplotlib: None
openpyxl: None
xlrd: None
xlwt: None
xlsxwriter: None
lxml: None
bs4: None
html5lib: 0.9999999
sqlalchemy: 1.1.9
pymysql: None
psycopg2: 2.7.1 (dt dec pq3 ext lo64)
jinja2: 2.9.5
s3fs: None
pandas_gbq: None
pandas_datareader: None

</details>
